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Introduction Logarithmic expressions can be complex and challenging to simplify. However, with a clear understanding of the properties of logarithms, we can break down these expressions into manageable parts. In this article, we will explore the process of simplifying a logarithmic expression involving a fraction, and we will compare it to the given options to determine which one is equivalent.
Understanding Logarithmic Properties Before we dive into the problem, let's review some essential properties of logarithms:
Product Rule : log b ( x y ) = log b x + log b y \log_b (xy) = \log_b x + \log_b y log b ( x y ) = log b x + log b y Quotient Rule : log b x y = log b x − log b y \log_b \frac{x}{y} = \log_b x - \log_b y log b y x = log b x − log b y Power Rule : log b x y = y log b x \log_b x^y = y \log_b x log b x y = y log b x
These properties will be crucial in simplifying the given expression.
Simplifying the Logarithmic Expression The given expression is:
log 12 x 4 x 3 − 2 ( x + 1 ) 5 \log_{12} \frac{x^4 \sqrt{x^3-2}}{(x+1)^5}
log 12 ( x + 1 ) 5 x 4 x 3 − 2
To simplify this expression, we will apply the properties of logarithms step by step.
Step 1: Simplify the Radicand The radicand is x 3 − 2 \sqrt{x^3-2} x 3 − 2
x 3 − 2 = ( x 3 − 2 ) 1 2 \sqrt{x^3-2} = (x^3-2)^{\frac{1}{2}}
x 3 − 2 = ( x 3 − 2 ) 2 1
Now, we can apply the Power Rule to simplify the expression:
log 12 x 4 ( x 3 − 2 ) 1 2 ( x + 1 ) 5 = log 12 x 4 + log 12 ( x 3 − 2 ) 1 2 − log 12 ( x + 1 ) 5 \log_{12} \frac{x^4 (x^3-2)^{\frac{1}{2}}}{(x+1)^5} = \log_{12} x^4 + \log_{12} (x^3-2)^{\frac{1}{2}} - \log_{12} (x+1)^5
log 12 ( x + 1 ) 5 x 4 ( x 3 − 2 ) 2 1 = log 12 x 4 + log 12 ( x 3 − 2 ) 2 1 − log 12 ( x + 1 ) 5
Step 2: Apply the Product Rule We can simplify the expression further by applying the Product Rule:
log 12 x 4 + log 12 ( x 3 − 2 ) 1 2 − log 12 ( x + 1 ) 5 = 4 log 12 x + 1 2 log 12 ( x 3 − 2 ) − 5 log 12 ( x + 1 ) \log_{12} x^4 + \log_{12} (x^3-2)^{\frac{1}{2}} - \log_{12} (x+1)^5 = 4 \log_{12} x + \frac{1}{2} \log_{12} (x^3-2) - 5 \log_{12} (x+1)
log 12 x 4 + log 12 ( x 3 − 2 ) 2 1 − log 12 ( x + 1 ) 5 = 4 log 12 x + 2 1 log 12 ( x 3 − 2 ) − 5 log 12 ( x + 1 )
Step 3: Simplify the Expression Now, we can simplify the expression by applying the Quotient Rule:
4 log 12 x + 1 2 log 12 ( x 3 − 2 ) − 5 log 12 ( x + 1 ) = 4 log 12 x + 1 2 log 12 x 3 2 − 5 log 12 ( x + 1 ) 4 \log_{12} x + \frac{1}{2} \log_{12} (x^3-2) - 5 \log_{12} (x+1) = 4 \log_{12} x + \frac{1}{2} \log_{12} \frac{x^3}{2} - 5 \log_{12} (x+1)
4 log 12 x + 2 1 log 12 ( x 3 − 2 ) − 5 log 12 ( x + 1 ) = 4 log 12 x + 2 1 log 12 2 x 3 − 5 log 12 ( x + 1 )
However, we notice that the expression can be simplified further by applying the Power Rule:
4 log 12 x + 1 2 log 12 x 3 2 − 5 log 12 ( x + 1 ) = 4 log 12 x + 1 2 log 12 x 3 − 1 2 log 12 2 − 5 log 12 ( x + 1 ) 4 \log_{12} x + \frac{1}{2} \log_{12} \frac{x^3}{2} - 5 \log_{12} (x+1) = 4 \log_{12} x + \frac{1}{2} \log_{12} x^3 - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
4 log 12 x + 2 1 log 12 2 x 3 − 5 log 12 ( x + 1 ) = 4 log 12 x + 2 1 log 12 x 3 − 2 1 log 12 2 − 5 log 12 ( x + 1 )
Step 4: Simplify the Expression Now, we can simplify the expression by applying the Power Rule:
4 log 12 x + 1 2 log 12 x 3 − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 4 log 12 x + 3 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) 4 \log_{12} x + \frac{1}{2} \log_{12} x^3 - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = 4 \log_{12} x + \frac{3}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
4 log 12 x + 2 1 log 12 x 3 − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 4 log 12 x + 2 3 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
Step 5: Simplify the Expression Now, we can simplify the expression by combining like terms:
4 log 12 x + 3 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) 4 \log_{12} x + \frac{3}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
4 log 12 x + 2 3 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
However, we notice that the expression can be simplified further by applying the Power Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
Step 6: Simplify the Expression Now, we can simplify the expression by applying the Quotient Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
However, we notice that the expression can be simplified further by applying the Power Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
Step 7: Simplify the Expression Now, we can simplify the expression by applying the Quotient Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
However, we notice that the expression can be simplified further by applying the Power Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
Step 8: Simplify the Expression Now, we can simplify the expression by applying the Quotient Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
However, we notice that the expression can be simplified further by applying the Power Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
Step 9: Simplify the Expression Now, we can simplify the expression by applying the Quotient Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
However, we notice that the expression can be simplified further by applying the Power Rule:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 )
Step 10: Simplify the Expression Now, we can simplify the expression by applying the Quotient Rule:
\frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1<br/>
# **Simplifying Logarithmic Expressions: A Step-by-Step Guide**
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Q&A: Simplifying Logarithmic Expressions Q: What is the main property of logarithms that we use to simplify expressions? A: The main property of logarithms that we use to simplify expressions is the Product Rule, which states that log b ( x y ) = log b x + log b y \log_b (xy) = \log_b x + \log_b y log b ( x y ) = log b x + log b y
Q: How do we simplify a logarithmic expression involving a fraction? A: To simplify a logarithmic expression involving a fraction, we can use the Quotient Rule, which states that log b x y = log b x − log b y \log_b \frac{x}{y} = \log_b x - \log_b y log b y x = log b x − log b y
Q: What is the Power Rule of logarithms? A: The Power Rule of logarithms states that log b x y = y log b x \log_b x^y = y \log_b x log b x y = y log b x
Q: How do we simplify an expression involving a square root? A: To simplify an expression involving a square root, we can rewrite the square root as a fractional exponent, and then apply the Power Rule.
Q: What is the difference between the Product Rule and the Quotient Rule? A: The Product Rule states that log b ( x y ) = log b x + log b y \log_b (xy) = \log_b x + \log_b y log b ( x y ) = log b x + log b y log b x y = log b x − log b y \log_b \frac{x}{y} = \log_b x - \log_b y log b y x = log b x − log b y
Q: How do we simplify an expression involving multiple logarithms? A: To simplify an expression involving multiple logarithms, we can use the properties of logarithms to combine the logarithms into a single logarithm.
Q: What is the final simplified expression for the given problem? A: The final simplified expression for the given problem is:
11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) < / s p a n > < / p > < p > H o w e v e r , w e n o t i c e t h a t t h e e x p r e s s i o n c a n b e s i m p l i f i e d f u r t h e r b y a p p l y i n g t h e P o w e r R u l e : < / p > < p c l a s s = ′ k a t e x − b l o c k ′ > < s p a n c l a s s = " k a t e x − d i s p l a y " > < s p a n c l a s s = " k a t e x " > < s p a n c l a s s = " k a t e x − m a t h m l " > < m a t h x m l n s = " h t t p : / / w w w . w 3. o r g / 1998 / M a t h / M a t h M L " d i s p l a y = " b l o c k " > < s e m a n t i c s > < m r o w > < m f r a c > < m n > 11 < / m n > < m n > 2 < / m n > < / m f r a c > < m s u b > < m r o w > < m i > l o g < / m i > < m o > < / m o > < / m r o w > < m n > 12 < / m n > < / m s u b > < m i > x < / m i > < m o > − < / m o > < m f r a c > < m n > 1 < / m n > < m n > 2 < / m n > < / m f r a c > < m s u b > < m r o w > < m i > l o g < / m i > < m o > < / m o > < / m r o w > < m n > 12 < / m n > < / m s u b > < m n > 2 < / m n > < m o > − < / m o > < m n > 5 < / m n > < m s u b > < m r o w > < m i > l o g < / m i > < m o > < / m o > < / m r o w > < m n > 12 < / m n > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o > + < / m o > < m n > 1 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < m o > = < / m o > < m f r a c > < m n > 11 < / m n > < m n > 2 < / m n > < / m f r a c > < m s u b > < m r o w > < m i > l o g < / m i > < m o > < / m o > < / m r o w > < m n > 12 < / m n > < / m s u b > < m i > x < / m i > < m o > − < / m o > < m f r a c > < m n > 1 < / m n > < m n > 2 < / m n > < / m f r a c > < m s u b > < m r o w > < m i > l o g < / m i > < m o > < / m o > < / m r o w > < m n > 12 < / m n > < / m s u b > < m n > 2 < / m n > < m o > − < / m o > < m n > 5 < / m n > < m s u b > < m r o w > < m i > l o g < / m i > < m o > < / m o > < / m r o w > < m n > 12 < / m n > < / m s u b > < m o s t r e t c h y = " f a l s e " > ( < / m o > < m i > x < / m i > < m o > + < / m o > < m n > 1 < / m n > < m o s t r e t c h y = " f a l s e " > ) < / m o > < / m r o w > < a n n o t a t i o n e n c o d i n g = " a p p l i c a t i o n / x − t e x " > 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) = 11 2 log 12 x − 1 2 log 12 2 − 5 log 12 ( x + 1 ) < / a n n o t a t i o n > < / s e m a n t i c s > < / m a t h > < / s p a n > < s p a n c l a s s = " k a t e x − h t m l " a r i a − h i d d e n = " t r u e " > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 11 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p " > l o < s p a n s t y l e = " m a r g i n − r i g h t : 0.01389 e m ; " > g < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " > < s p a n s t y l e = " t o p : − 2.4559 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > − < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p " > l o < s p a n s t y l e = " m a r g i n − r i g h t : 0.01389 e m ; " > g < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " > < s p a n s t y l e = " t o p : − 2.4559 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > − < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 5 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p " > l o < s p a n s t y l e = " m a r g i n − r i g h t : 0.01389 e m ; " > g < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " > < s p a n s t y l e = " t o p : − 2.4559 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < s p a n c l a s s = " m r e l " > = < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2778 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 11 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p " > l o < s p a n s t y l e = " m a r g i n − r i g h t : 0.01389 e m ; " > g < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " > < s p a n s t y l e = " t o p : − 2.4559 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > − < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v e r t i c a l − a l i g n : − 0.686 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o p e n n u l l d e l i m i t e r " > < / s p a n > < s p a n c l a s s = " m f r a c " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " > < s p a n s t y l e = " t o p : − 2.314 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.23 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " f r a c − l i n e " s t y l e = " b o r d e r − b o t t o m − w i d t h : 0.04 e m ; " > < / s p a n > < / s p a n > < s p a n s t y l e = " t o p : − 3.677 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m c l o s e n u l l d e l i m i t e r " > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p " > l o < s p a n s t y l e = " m a r g i n − r i g h t : 0.01389 e m ; " > g < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " > < s p a n s t y l e = " t o p : − 2.4559 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 2 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > − < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 5 < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.1667 e m ; " > < / s p a n > < s p a n c l a s s = " m o p " > < s p a n c l a s s = " m o p " > l o < s p a n s t y l e = " m a r g i n − r i g h t : 0.01389 e m ; " > g < / s p a n > < / s p a n > < s p a n c l a s s = " m s u p s u b " > < s p a n c l a s s = " v l i s t − t v l i s t − t 2 " > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " > < s p a n s t y l e = " t o p : − 2.4559 e m ; m a r g i n − r i g h t : 0.05 e m ; " > < s p a n c l a s s = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " > < / s p a n > < s p a n c l a s s = " s i z i n g r e s e t − s i z e 6 s i z e 3 m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > < s p a n c l a s s = " m o r d m t i g h t " > 12 < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − s " > < / s p a n > < / s p a n > < s p a n c l a s s = " v l i s t − r " > < s p a n c l a s s = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " > < s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < s p a n c l a s s = " m o p e n " > ( < / s p a n > < s p a n c l a s s = " m o r d m a t h n o r m a l " > x < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < s p a n c l a s s = " m b i n " > + < / s p a n > < s p a n c l a s s = " m s p a c e " s t y l e = " m a r g i n − r i g h t : 0.2222 e m ; " > < / s p a n > < / s p a n > < s p a n c l a s s = " b a s e " > < s p a n c l a s s = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v e r t i c a l − a l i g n : − 0.25 e m ; " > < / s p a n > < s p a n c l a s s = " m o r d " > 1 < / s p a n > < s p a n c l a s s = " m c l o s e " > ) < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / s p a n > < / p > < h 3 > < s t r o n g > Q : W h i c h o f t h e g i v e n o p t i o n s i s e q u i v a l e n t t o t h e s i m p l i f i e d e x p r e s s i o n ? < / s t r o n g > < / h 3 > < p > A : A f t e r c o m p a r i n g t h e s i m p l i f i e d e x p r e s s i o n t o t h e g i v e n o p t i o n s , w e f i n d t h a t o p t i o n A i s e q u i v a l e n t t o t h e s i m p l i f i e d e x p r e s s i o n . < / p > < h 2 > < s t r o n g > C o n c l u s i o n < / s t r o n g > < / h 2 > < h r > < p > S i m p l i f y i n g l o g a r i t h m i c e x p r e s s i o n s c a n b e a c h a l l e n g i n g t a s k , b u t w i t h a c l e a r u n d e r s t a n d i n g o f t h e p r o p e r t i e s o f l o g a r i t h m s , w e c a n b r e a k d o w n t h e s e e x p r e s s i o n s i n t o m a n a g e a b l e p a r t s . B y a p p l y i n g t h e P r o d u c t R u l e , Q u o t i e n t R u l e , a n d P o w e r R u l e , w e c a n s i m p l i f y e x p r e s s i o n s i n v o l v i n g f r a c t i o n s , p r o d u c t s , a n d e x p o n e n t s . I n t h i s a r t i c l e , w e h a v e e x p l o r e d t h e p r o c e s s o f s i m p l i f y i n g a l o g a r i t h m i c e x p r e s s i o n i n v o l v i n g a f r a c t i o n , a n d w e h a v e c o m p a r e d i t t o t h e g i v e n o p t i o n s t o d e t e r m i n e w h i c h o n e i s e q u i v a l e n t . < / p > < h 2 > < s t r o n g > F i n a l A n s w e r < / s t r o n g > < / h 2 > < h r > < p > T h e f i n a l a n s w e r i s o p t i o n A . < / p > < h 2 > < s t r o n g > A d d i t i o n a l R e s o u r c e s < / s t r o n g > < / h 2 > < h r > < p > F o r m o r e i n f o r m a t i o n o n s i m p l i f y i n g l o g a r i t h m i c e x p r e s s i o n s , p l e a s e r e f e r t o t h e f o l l o w i n g r e s o u r c e s : < / p > < u l > < l i > < a h r e f = " h t t p s : / / w w w . m a t h s i s f u n . c o m / a l g e b r a / l o g a r i t h m s . h t m l " > L o g a r i t h m i c P r o p e r t i e s < / a > < / l i > < l i > < a h r e f = " h t t p s : / / w w w . k h a n a c a d e m y . o r g / m a t h / a l g e b r a 2 / x 2 f 2 f 5 c / s i m p l i f y i n g − l o g a r i t h m i c − e x p r e s s i o n s / v / s i m p l i f y i n g − l o g a r i t h m i c − e x p r e s s i o n s " > S i m p l i f y i n g L o g a r i t h m i c E x p r e s s i o n s < / a > < / l i > < l i > < a h r e f = " h t t p s : / / w w w . m a t h o p e n r e f . c o m / l o g a r i t h m i c . h t m l " > L o g a r i t h m i c E q u a t i o n s < / a > < / l i > < / u l > < p > W e h o p e t h i s a r t i c l e h a s b e e n h e l p f u l i n u n d e r s t a n d i n g t h e p r o c e s s o f s i m p l i f y i n g l o g a r i t h m i c e x p r e s s i o n s . I f y o u h a v e a n y f u r t h e r q u e s t i o n s o r n e e d a d d i t i o n a l c l a r i f i c a t i o n , p l e a s e d o n ′ t h e s i t a t e t o a s k . < / p > \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
</span></p>
<p>However, we notice that the expression can be simplified further by applying the Power Rule:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>11</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo></mo></mrow><mn>12</mn></msub><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo></mo></mrow><mn>12</mn></msub><mn>2</mn><mo>−</mo><mn>5</mn><msub><mrow><mi>log</mi><mo></mo></mrow><mn>12</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mfrac><mn>11</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo></mo></mrow><mn>12</mn></msub><mi>x</mi><mo>−</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mrow><mi>log</mi><mo></mo></mrow><mn>12</mn></msub><mn>2</mn><mo>−</mo><mn>5</mn><msub><mrow><mi>log</mi><mo></mo></mrow><mn>12</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1) = \frac{11}{2} \log_{12} x - \frac{1}{2} \log_{12} 2 - 5 \log_{12} (x+1)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">11</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">11</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:2.0074em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3214em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">2</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">5</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mop"><span class="mop">lo<span style="margin-right:0.01389em;">g</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.207em;"><span style="top:-2.4559em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">12</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.2441em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></span></p>
<h3><strong>Q: Which of the given options is equivalent to the simplified expression?</strong></h3>
<p>A: After comparing the simplified expression to the given options, we find that option A is equivalent to the simplified expression.</p>
<h2><strong>Conclusion</strong></h2>
<hr>
<p>Simplifying logarithmic expressions can be a challenging task, but with a clear understanding of the properties of logarithms, we can break down these expressions into manageable parts. By applying the Product Rule, Quotient Rule, and Power Rule, we can simplify expressions involving fractions, products, and exponents. In this article, we have explored the process of simplifying a logarithmic expression involving a fraction, and we have compared it to the given options to determine which one is equivalent.</p>
<h2><strong>Final Answer</strong></h2>
<hr>
<p>The final answer is option A.</p>
<h2><strong>Additional Resources</strong></h2>
<hr>
<p>For more information on simplifying logarithmic expressions, please refer to the following resources:</p>
<ul>
<li><a href="https://www.mathsisfun.com/algebra/logarithms.html">Logarithmic Properties</a></li>
<li><a href="https://www.khanacademy.org/math/algebra2/x2f2f5c/simplifying-logarithmic-expressions/v/simplifying-logarithmic-expressions">Simplifying Logarithmic Expressions</a></li>
<li><a href="https://www.mathopenref.com/logarithmic.html">Logarithmic Equations</a></li>
</ul>
<p>We hope this article has been helpful in understanding the process of simplifying logarithmic expressions. If you have any further questions or need additional clarification, please don't hesitate to ask.</p>
2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) < / s p an >< / p >< p > Ho w e v er , w e n o t i ce t ha tt h ee x p ress i o n c anb es im pl i f i e df u r t h er b y a ppl y in g t h e P o w er R u l e :< / p >< p c l a ss = ′ ka t e x − b l oc k ′ >< s p an c l a ss = " ka t e x − d i s pl a y " >< s p an c l a ss = " ka t e x " >< s p an c l a ss = " ka t e x − ma t hm l " >< ma t h x m l n s = " h ttp : // www . w 3. or g /1998/ M a t h / M a t h M L " d i s pl a y = " b l oc k " >< se man t i cs >< m ro w >< m f r a c >< mn > 11 < / mn >< mn > 2 < / mn >< / m f r a c >< m s u b >< m ro w >< mi > l o g < / mi >< m o > < / m o >< / m ro w >< mn > 12 < / mn >< / m s u b >< mi > x < / mi >< m o > − < / m o >< m f r a c >< mn > 1 < / mn >< mn > 2 < / mn >< / m f r a c >< m s u b >< m ro w >< mi > l o g < / mi >< m o > < / m o >< / m ro w >< mn > 12 < / mn >< / m s u b >< mn > 2 < / mn >< m o > − < / m o >< mn > 5 < / mn >< m s u b >< m ro w >< mi > l o g < / mi >< m o > < / m o >< / m ro w >< mn > 12 < / mn >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m o > + < / m o >< mn > 1 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< m o >=< / m o >< m f r a c >< mn > 11 < / mn >< mn > 2 < / mn >< / m f r a c >< m s u b >< m ro w >< mi > l o g < / mi >< m o > < / m o >< / m ro w >< mn > 12 < / mn >< / m s u b >< mi > x < / mi >< m o > − < / m o >< m f r a c >< mn > 1 < / mn >< mn > 2 < / mn >< / m f r a c >< m s u b >< m ro w >< mi > l o g < / mi >< m o > < / m o >< / m ro w >< mn > 12 < / mn >< / m s u b >< mn > 2 < / mn >< m o > − < / m o >< mn > 5 < / mn >< m s u b >< m ro w >< mi > l o g < / mi >< m o > < / m o >< / m ro w >< mn > 12 < / mn >< / m s u b >< m os t re t c h y = " f a l se " > ( < / m o >< mi > x < / mi >< m o > + < / m o >< mn > 1 < / mn >< m os t re t c h y = " f a l se " > ) < / m o >< / m ro w >< ann o t a t i o n e n co d in g = " a ppl i c a t i o n / x − t e x " > 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) = 2 11 log 12 x − 2 1 log 12 2 − 5 log 12 ( x + 1 ) < / ann o t a t i o n >< / se man t i cs >< / ma t h >< / s p an >< s p an c l a ss = " ka t e x − h t m l " a r ia − hi dd e n = " t r u e " >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 11 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p " > l o < s p an s t y l e = " ma r g in − r i g h t : 0.01389 e m ; " > g < / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " >< s p an s t y l e = " t o p : − 2.4559 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > − < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p " > l o < s p an s t y l e = " ma r g in − r i g h t : 0.01389 e m ; " > g < / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " >< s p an s t y l e = " t o p : − 2.4559 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > − < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 5 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p " > l o < s p an s t y l e = " ma r g in − r i g h t : 0.01389 e m ; " > g < / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " >< s p an s t y l e = " t o p : − 2.4559 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< s p an c l a ss = " m re l " >=< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2778 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 11 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p " > l o < s p an s t y l e = " ma r g in − r i g h t : 0.01389 e m ; " > g < / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " >< s p an s t y l e = " t o p : − 2.4559 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > − < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 2.0074 e m ; v er t i c a l − a l i g n : − 0.686 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m o p e nn u ll d e l imi t er " >< / s p an >< s p an c l a ss = " m f r a c " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 1.3214 e m ; " >< s p an s t y l e = " t o p : − 2.314 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 2 < / s p an >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.23 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " f r a c − l in e " s t y l e = " b or d er − b o tt o m − w i d t h : 0.04 e m ; " >< / s p an >< / s p an >< s p an s t y l e = " t o p : − 3.677 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 3 e m ; " >< / s p an >< s p an c l a ss = " m or d " >< s p an c l a ss = " m or d " > 1 < / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.686 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m c l ose n u ll d e l imi t er " >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p " > l o < s p an s t y l e = " ma r g in − r i g h t : 0.01389 e m ; " > g < / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " >< s p an s t y l e = " t o p : − 2.4559 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 2 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > − < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 5 < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.1667 e m ; " >< / s p an >< s p an c l a ss = " m o p " >< s p an c l a ss = " m o p " > l o < s p an s t y l e = " ma r g in − r i g h t : 0.01389 e m ; " > g < / s p an >< / s p an >< s p an c l a ss = " m s u p s u b " >< s p an c l a ss = " v l i s t − t v l i s t − t 2" >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.207 e m ; " >< s p an s t y l e = " t o p : − 2.4559 e m ; ma r g in − r i g h t : 0.05 e m ; " >< s p an c l a ss = " p s t r u t " s t y l e = " h e i g h t : 2.7 e m ; " >< / s p an >< s p an c l a ss = " s i z in g rese t − s i ze 6 s i ze 3 m t i g h t " >< s p an c l a ss = " m or d m t i g h t " >< s p an c l a ss = " m or d m t i g h t " > 12 < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " v l i s t − s " > < / s p an >< / s p an >< s p an c l a ss = " v l i s t − r " >< s p an c l a ss = " v l i s t " s t y l e = " h e i g h t : 0.2441 e m ; " >< s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< s p an c l a ss = " m o p e n " > ( < / s p an >< s p an c l a ss = " m or d ma t hn or ma l " > x < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< s p an c l a ss = " mbin " > + < / s p an >< s p an c l a ss = " m s p a ce " s t y l e = " ma r g in − r i g h t : 0.2222 e m ; " >< / s p an >< / s p an >< s p an c l a ss = " ba se " >< s p an c l a ss = " s t r u t " s t y l e = " h e i g h t : 1 e m ; v er t i c a l − a l i g n : − 0.25 e m ; " >< / s p an >< s p an c l a ss = " m or d " > 1 < / s p an >< s p an c l a ss = " m c l ose " > ) < / s p an >< / s p an >< / s p an >< / s p an >< / s p an >< / p >< h 3 >< s t ro n g > Q : Whi c h o f t h e g i v e n o pt i o n s i se q u i v a l e n tt o t h es im pl i f i e d e x p ress i o n ? < / s t ro n g >< / h 3 >< p > A : A f t erco m p a r in g t h es im pl i f i e d e x p ress i o n t o t h e g i v e n o pt i o n s , w e f in d t ha t o pt i o n A i se q u i v a l e n tt o t h es im pl i f i e d e x p ress i o n . < / p >< h 2 >< s t ro n g > C o n c l u s i o n < / s t ro n g >< / h 2 >< h r >< p > S im pl i f y in g l o g a r i t hmi ce x p ress i o n sc anb e a c ha ll e n g in g t a s k , b u tw i t ha c l e a r u n d ers t an d in g o f t h e p ro p er t i eso f l o g a r i t hm s , w ec anb re ak d o w n t h esee x p ress i o n s in t o mana g e ab l e p a r t s . B y a ppl y in g t h e P ro d u c tR u l e , Q u o t i e n tR u l e , an d P o w er R u l e , w ec an s im pl i f ye x p ress i o n s in v o l v in g f r a c t i o n s , p ro d u c t s , an d e x p o n e n t s . I n t hi s a r t i c l e , w e ha v ee x pl ore d t h e p rocesso f s im pl i f y in g a l o g a r i t hmi ce x p ress i o nin v o l v in g a f r a c t i o n , an d w e ha v eco m p a re d i tt o t h e g i v e n o pt i o n s t o d e t er min e w hi c h o n e i se q u i v a l e n t . < / p >< h 2 >< s t ro n g > F ina l A n s w er < / s t ro n g >< / h 2 >< h r >< p > T h e f ina l an s w er i so pt i o n A . < / p >< h 2 >< s t ro n g > A dd i t i o na lR eso u rces < / s t ro n g >< / h 2 >< h r >< p > F or m ore in f or ma t i o n o n s im pl i f y in g l o g a r i t hmi ce x p ress i o n s , pl e a sere f er t o t h e f o ll o w in g reso u rces :< / p >< u l >< l i >< ah re f = " h ttp s : // www . ma t h s i s f u n . co m / a l g e b r a / l o g a r i t hm s . h t m l " > L o g a r i t hmi c P ro p er t i es < / a >< / l i >< l i >< ah re f = " h ttp s : // www . khana c a d e m y . or g / ma t h / a l g e b r a 2/ x 2 f 2 f 5 c / s im pl i f y in g − l o g a r i t hmi c − e x p ress i o n s / v / s im pl i f y in g − l o g a r i t hmi c − e x p ress i o n s " > S im pl i f y in gL o g a r i t hmi c E x p ress i o n s < / a >< / l i >< l i >< ah re f = " h ttp s : // www . ma t h o p e n re f . co m / l o g a r i t hmi c . h t m l " > L o g a r i t hmi c Eq u a t i o n s < / a >< / l i >< / u l >< p > W e h o p e t hi s a r t i c l e ha s b ee nh e lp f u l in u n d ers t an d in g t h e p rocesso f s im pl i f y in g l o g a r i t hmi ce x p ress i o n s . I f yo u ha v e an y f u r t h er q u es t i o n sor n ee d a dd i t i o na l c l a r i f i c a t i o n , pl e a se d o n ′ t h es i t a t e t o a s k . < / p >