Simplify The Following Into A Single Logarithm:A. 3 Log ⁡ ( 4 ) + 2 Log ⁡ ( X 3 \log (4) + 2 \log (x 3 Lo G ( 4 ) + 2 Lo G ( X ]B. \log \left(3 \cdot 4 \cdot X^2\right ]C. Log ⁡ ( 3 ⋅ 4 ⋅ 2 X \log (3 \cdot 4 \cdot 2 X Lo G ( 3 ⋅ 4 ⋅ 2 X ]D. \log \left(\frac{4^3}{x^2}\right ]E. $\log \left(4^3

Introduction

Logarithms are a fundamental concept in mathematics, and they play a crucial role in various mathematical operations. In this article, we will explore how to simplify the given expressions into a single logarithm. We will use the properties of logarithms to combine the terms and express them in a simpler form.

Property 1: Simplifying the Sum of Logarithms

The first expression we need to simplify is:

$3 \log (4) + 2 \log (x)$

To simplify this expression, we can use the property of logarithms that states:

$\log (a) + \log (b) = \log (a \cdot b)$

Using this property, we can rewrite the expression as:

$3 \log (4) + 2 \log (x) = \log (4^3) + \log (x^2)$

Now, we can use the property of logarithms that states:

$\log (a) + \log (b) = \log (a \cdot b)$

to combine the two logarithmic terms:

$\log (4^3) + \log (x^2) = \log (4^3 \cdot x^2)$

Therefore, the simplified expression is:

$\log (4^3 \cdot x^2)$

Property 2: Simplifying the Product of Logarithms

The second expression we need to simplify is:

$\log \left(3 \cdot 4 \cdot x^2\right)$

To simplify this expression, we can use the property of logarithms that states:

$\log (a \cdot b) = \log (a) + \log (b)$

Using this property, we can rewrite the expression as:

$\log \left(3 \cdot 4 \cdot x^2\right) = \log (3) + \log (4) + \log (x^2)$

Now, we can use the property of logarithms that states:

$\log (a) + \log (b) = \log (a \cdot b)$

to combine the three logarithmic terms:

$\log (3) + \log (4) + \log (x^2) = \log (3 \cdot 4 \cdot x^2)$

Therefore, the simplified expression is:

$\log (3 \cdot 4 \cdot x^2)$

Property 3: Simplifying the Quotient of Logarithms

The third expression we need to simplify is:

$\log (3 \cdot 4 \cdot 2 x)$

To simplify this expression, we can use the property of logarithms that states:

$\log (a \cdot b) = \log (a) + \log (b)$

Using this property, we can rewrite the expression as:

$\log (3 \cdot 4 \cdot 2 x) = \log (3) + \log (4) + \log (2 x)$

Now, we can use the property of logarithms that states:

$\log (a) + \log (b) = \log (a \cdot b)$

to combine the three logarithmic terms:

$\log (3) + \log (4) + \log (2 x) = \log (3 \cdot 4 \cdot 2 x)$

Therefore, the simplified expression is:

$\log (3 \cdot 4 \cdot 2 x)$

Property 4: Simplifying the Quotient of Logarithms

The fourth expression we need to simplify is:

$\log \left(\frac{4^3}{x^2}\right)$

To simplify this expression, we can use the property of logarithms that states:

$\log \left(\frac{a}{b}\right) = \log (a) - \log (b)$

Using this property, we can rewrite the expression as:

$\log \left(\frac{4^3}{x^2}\right) = \log (4^3) - \log (x^2)$

Now, we can use the property of logarithms that states:

$\log (a) - \log (b) = \log \left(\frac{a}{b}\right)$

to simplify the expression:

$\log (4^3) - \log (x^2) = \log \left(\frac{4^3}{x^2}\right)$

Therefore, the simplified expression is:

$\log \left(\frac{4^3}{x^2}\right)$

Property 5: Simplifying the Power of a Logarithm

The fifth expression we need to simplify is:

$\log \left(4^3\right)$

To simplify this expression, we can use the property of logarithms that states:

$\log (a^b) = b \log (a)$

Using this property, we can rewrite the expression as:

$\log \left(4^3\right) = 3 \log (4)$

Therefore, the simplified expression is:

$3 \log (4)$

Conclusion

In this article, we have explored how to simplify the given expressions into a single logarithm. We have used the properties of logarithms to combine the terms and express them in a simpler form. The properties of logarithms that we have used include the sum of logarithms, the product of logarithms, the quotient of logarithms, and the power of a logarithm. By applying these properties, we have been able to simplify the expressions and express them in a more compact form.

Final Answer

The final answers to the given expressions are:

A. $\log (4^3 \cdot x^2)$

B. $\log (3 \cdot 4 \cdot x^2)$

C. $\log (3 \cdot 4 \cdot 2 x)$

D. $\log \left(\frac{4^3}{x^2}\right)$

E. $3 \log (4)$

Introduction

In our previous article, we explored how to simplify the given expressions into a single logarithm. We used the properties of logarithms to combine the terms and express them in a simpler form. In this article, we will answer some frequently asked questions related to simplifying logarithmic expressions.

Q1: What is the property of logarithms that states $\log (a) + \log (b) = \log (a \cdot b)$?

A1: This property is known as the sum of logarithms. It states that the sum of two logarithmic terms with the same base is equal to the logarithm of their product.

Q2: How do I simplify the expression $\log (3 \cdot 4 \cdot x^2)$ using the sum of logarithms property?

A2: To simplify this expression, you can use the sum of logarithms property to rewrite it as:

$\log (3 \cdot 4 \cdot x^2) = \log (3) + \log (4) + \log (x^2)$

Then, you can use the product of logarithms property to combine the three logarithmic terms:

$\log (3) + \log (4) + \log (x^2) = \log (3 \cdot 4 \cdot x^2)$

Q3: What is the property of logarithms that states $\log (a \cdot b) = \log (a) + \log (b)$?

A3: This property is known as the product of logarithms. It states that the logarithm of a product is equal to the sum of the logarithms of its factors.

Q4: How do I simplify the expression $\log \left(\frac{4^3}{x^2}\right)$ using the quotient of logarithms property?

A4: To simplify this expression, you can use the quotient of logarithms property to rewrite it as:

$\log \left(\frac{4^3}{x^2}\right) = \log (4^3) - \log (x^2)$

Then, you can use the power of a logarithm property to simplify the expression:

$\log (4^3) - \log (x^2) = 3 \log (4) - \log (x^2)$

Q5: What is the property of logarithms that states $\log (a^b) = b \log (a)$?

A5: This property is known as the power of a logarithm. It states that the logarithm of a power is equal to the exponent multiplied by the logarithm of the base.

Q6: How do I simplify the expression $\log (4^3)$ using the power of a logarithm property?

A6: To simplify this expression, you can use the power of a logarithm property to rewrite it as:

$\log (4^3) = 3 \log (4)$

Q7: What is the difference between the sum of logarithms and the product of logarithms properties?

A7: The sum of logarithms property states that the sum of two logarithmic terms with the same base is equal to the logarithm of their product. The product of logarithms property states that the logarithm of a product is equal to the sum of the logarithms of its factors.

Q8: How do I use the properties of logarithms to simplify a logarithmic expression?

A8: To simplify a logarithmic expression, you can use the properties of logarithms to combine the terms and express them in a simpler form. You can use the sum of logarithms property to combine two or more logarithmic terms with the same base, the product of logarithms property to combine the logarithms of a product, and the quotient of logarithms property to combine the logarithms of a quotient.

Conclusion

In this article, we have answered some frequently asked questions related to simplifying logarithmic expressions. We have used the properties of logarithms to combine the terms and express them in a simpler form. By applying these properties, we have been able to simplify the expressions and express them in a more compact form.

Final Answer

The final answers to the given questions are:

Q1: $\log (a) + \log (b) = \log (a \cdot b)$

Q2: $\log (3 \cdot 4 \cdot x^2) = \log (3) + \log (4) + \log (x^2)$

Q3: $\log (a \cdot b) = \log (a) + \log (b)$

Q4: $\log \left(\frac{4^3}{x^2}\right) = 3 \log (4) - \log (x^2)$

Q5: $\log (a^b) = b \log (a)$

Q6: $\log (4^3) = 3 \log (4)$

Q7: The sum of logarithms property states that the sum of two logarithmic terms with the same base is equal to the logarithm of their product. The product of logarithms property states that the logarithm of a product is equal to the sum of the logarithms of its factors.

Q8: To simplify a logarithmic expression, you can use the properties of logarithms to combine the terms and express them in a simpler form. You can use the sum of logarithms property to combine two or more logarithmic terms with the same base, the product of logarithms property to combine the logarithms of a product, and the quotient of logarithms property to combine the logarithms of a quotient.