The Expression Log ⁡ 1 3 Log ⁡ 2 \frac{\log \frac{1}{3}}{\log 2} L O G 2 L O G 3 1 ​ ​ Is The Result Of Applying The Change Of Base Formula To A Logarithmic Expression. Which Could Be The Original Expression?A. Log ⁡ 1 3 2 \log _{\frac{1}{3}} 2 Lo G 3 1 ​ ​ 2 B. Log ⁡ 1 2 3 \log _{\frac{1}{2}} 3 Lo G 2 1 ​ ​ 3

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The Expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}: Unraveling the Original Logarithmic Expression

The change of base formula is a fundamental concept in mathematics, particularly in the realm of logarithms. It allows us to express a logarithmic expression in terms of a different base, making it easier to work with and manipulate. In this article, we will delve into the expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2} and explore which original logarithmic expression could have resulted in this outcome.

Understanding the Change of Base Formula

The change of base formula states that for any positive real numbers aa, bb, and cc, where c≠1c \neq 1, the following equation holds:

log⁡ab=log⁡cblog⁡ca\log_a b = \frac{\log_c b}{\log_c a}

This formula enables us to change the base of a logarithmic expression from aa to cc, making it a powerful tool for simplifying and solving logarithmic equations.

Applying the Change of Base Formula

Given the expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}, we can apply the change of base formula to identify the original logarithmic expression. Let's start by examining the numerator, log⁡13\log \frac{1}{3}. We can rewrite this expression using the properties of logarithms:

log⁡13=log⁡1−log⁡3=0−log⁡3=−log⁡3\log \frac{1}{3} = \log 1 - \log 3 = 0 - \log 3 = -\log 3

Now, let's focus on the denominator, log⁡2\log 2. This expression remains unchanged.

Original Expression: log⁡132\log _{\frac{1}{3}} 2 or log⁡123\log _{\frac{1}{2}} 3?

We are given two possible original expressions: log⁡132\log _{\frac{1}{3}} 2 and log⁡123\log _{\frac{1}{2}} 3. Let's analyze each expression separately to determine which one could have resulted in the expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}.

Option A: log⁡132\log _{\frac{1}{3}} 2

Using the change of base formula, we can rewrite the expression log⁡132\log _{\frac{1}{3}} 2 as:

log⁡132=log⁡2log⁡13\log _{\frac{1}{3}} 2 = \frac{\log 2}{\log \frac{1}{3}}

Now, let's substitute the expression log⁡13=−log⁡3\log \frac{1}{3} = -\log 3 into the denominator:

log⁡132=log⁡2−log⁡3=−log⁡2log⁡3\log _{\frac{1}{3}} 2 = \frac{\log 2}{-\log 3} = -\frac{\log 2}{\log 3}

This expression is equivalent to the original expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}, but with a negative sign. However, the negative sign can be eliminated by multiplying both the numerator and denominator by −1-1:

−log⁡2log⁡3=−log⁡2log⁡3=log⁡13log⁡2-\frac{\log 2}{\log 3} = \frac{-\log 2}{\log 3} = \frac{\log \frac{1}{3}}{\log 2}

Therefore, the original expression log⁡132\log _{\frac{1}{3}} 2 could have resulted in the expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}.

Option B: log⁡123\log _{\frac{1}{2}} 3

Using the change of base formula, we can rewrite the expression log⁡123\log _{\frac{1}{2}} 3 as:

log⁡123=log⁡3log⁡12\log _{\frac{1}{2}} 3 = \frac{\log 3}{\log \frac{1}{2}}

Now, let's substitute the expression log⁡12=−log⁡2\log \frac{1}{2} = -\log 2 into the denominator:

log⁡123=log⁡3−log⁡2=−log⁡3log⁡2\log _{\frac{1}{2}} 3 = \frac{\log 3}{-\log 2} = -\frac{\log 3}{\log 2}

This expression is not equivalent to the original expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}. Therefore, the original expression log⁡123\log _{\frac{1}{2}} 3 could not have resulted in the expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}.

In conclusion, the original expression log⁡132\log _{\frac{1}{3}} 2 could have resulted in the expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2} using the change of base formula. This demonstrates the power of the change of base formula in simplifying and solving logarithmic equations.

The change of base formula is a fundamental concept in mathematics, and understanding its application is crucial for solving logarithmic equations. By applying the change of base formula, we can simplify complex logarithmic expressions and identify the original expression that resulted in a given expression. In this article, we explored the expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2} and determined that the original expression log⁡132\log _{\frac{1}{3}} 2 could have resulted in this outcome.
Frequently Asked Questions: The Change of Base Formula and Logarithmic Expressions

The change of base formula is a fundamental concept in mathematics, particularly in the realm of logarithms. It allows us to express a logarithmic expression in terms of a different base, making it easier to work with and manipulate. In this article, we will address some of the most frequently asked questions related to the change of base formula and logarithmic expressions.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to change the base of a logarithmic expression from one base to another. It is given by the equation:

log⁡ab=log⁡cblog⁡ca\log_a b = \frac{\log_c b}{\log_c a}

where aa, bb, and cc are positive real numbers, and c≠1c \neq 1.

Q: How do I apply the change of base formula?

A: To apply the change of base formula, you need to identify the base of the logarithmic expression you want to change. Then, you need to choose a new base and substitute the expressions into the formula. For example, if you want to change the base of the logarithmic expression log⁡23\log_2 3 to base 1010, you would use the formula:

log⁡23=log⁡103log⁡102\log_2 3 = \frac{\log_{10} 3}{\log_{10} 2}

Q: What are some common applications of the change of base formula?

A: The change of base formula has many applications in mathematics, particularly in the realm of logarithms. Some common applications include:

  • Simplifying complex logarithmic expressions
  • Changing the base of a logarithmic expression
  • Solving logarithmic equations
  • Finding the value of a logarithmic expression

Q: Can I use the change of base formula to change the base of a logarithmic expression with a negative exponent?

A: Yes, you can use the change of base formula to change the base of a logarithmic expression with a negative exponent. However, you need to be careful when dealing with negative exponents, as they can change the sign of the expression.

Q: How do I deal with logarithmic expressions with a base of 1?

A: Logarithmic expressions with a base of 1 are undefined, as the logarithm of 1 is 0. Therefore, you cannot use the change of base formula to change the base of a logarithmic expression with a base of 1.

Q: Can I use the change of base formula to change the base of a logarithmic expression with a fractional base?

A: Yes, you can use the change of base formula to change the base of a logarithmic expression with a fractional base. However, you need to be careful when dealing with fractional bases, as they can lead to complex expressions.

Q: What are some common mistakes to avoid when using the change of base formula?

A: Some common mistakes to avoid when using the change of base formula include:

  • Forgetting to change the base of the logarithmic expression
  • Not substituting the expressions into the formula correctly
  • Not simplifying the expression after applying the change of base formula
  • Not checking for undefined expressions

In conclusion, the change of base formula is a powerful tool for simplifying and solving logarithmic expressions. By understanding how to apply the change of base formula, you can simplify complex logarithmic expressions and solve logarithmic equations. Remember to be careful when dealing with negative exponents, fractional bases, and undefined expressions.

The change of base formula is a fundamental concept in mathematics, and understanding its application is crucial for solving logarithmic equations. By practicing the change of base formula and applying it to different types of logarithmic expressions, you can become proficient in using this powerful tool.