The Expression $\frac{\log \frac{1}{3}}{\log 2}$ Is The Result Of Applying The Change Of Base Formula To A Logarithmic Expression. Which Could Be The Original Expression?A. $\log _{\frac{1}{3}} 2$B. $\log _{\frac{1}{2}} 3$C.

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Introduction

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 change of base formula and explore how it can be applied to a given logarithmic expression. We will also examine the possible original expressions that could result in the given expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}.

Understanding the Change of Base Formula

The change of base formula states that for any positive real numbers a,b>1a, b > 1 and x>0x > 0, the following equation holds:

log⁡ax=log⁡bxlog⁡ba\log_a x = \frac{\log_b x}{\log_b a}

This formula allows us to express a logarithmic expression in terms of a different base. For example, if we want to express log⁡2x\log_2 x in terms of base 10, we can use the change of base formula:

log⁡2x=log⁡10xlog⁡102\log_2 x = \frac{\log_{10} x}{\log_{10} 2}

Applying the Change of Base Formula

Now, let's apply the change of base formula to the given expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}. We can rewrite the expression as:

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

Using the properties of logarithms, we can simplify the expression further:

log⁡1−log⁡3log⁡2=0−log⁡3log⁡2=−log⁡3log⁡2\frac{\log 1 - \log 3}{\log 2} = \frac{0 - \log 3}{\log 2} = -\frac{\log 3}{\log 2}

Now, we can apply the change of base formula to the expression:

−log⁡3log⁡2=−log⁡23log⁡22-\frac{\log 3}{\log 2} = -\frac{\log_2 3}{\log_2 2}

Possible Original Expressions

We are given three possible original expressions: log⁡132\log _{\frac{1}{3}} 2, log⁡123\log _{\frac{1}{2}} 3, and log⁡122\log _{\frac{1}{2}} 2. Let's examine each of these expressions and determine which one could result in the given expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}.

A. log⁡132\log _{\frac{1}{3}} 2

To determine if this expression could result in the given expression, we can apply the change of base formula:

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

This expression is not equal to the given expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}. Therefore, we can rule out option A.

B. log⁡123\log _{\frac{1}{2}} 3

To determine if this expression could result in the given expression, we can apply the change of base formula:

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

This expression is equal to the given expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}. Therefore, we can conclude that option B is the correct original expression.

C. log⁡122\log _{\frac{1}{2}} 2

To determine if this expression could result in the given expression, we can apply the change of base formula:

log⁡122=log⁡2log⁡12=log⁡2log⁡1−log⁡2=log⁡2−log⁡2=−1\log _{\frac{1}{2}} 2 = \frac{\log 2}{\log \frac{1}{2}} = \frac{\log 2}{\log 1 - \log 2} = \frac{\log 2}{- \log 2} = -1

This expression is not equal to the given expression log⁡13log⁡2\frac{\log \frac{1}{3}}{\log 2}. Therefore, we can rule out option C.

Conclusion

In this article, we applied the change of base formula to a given logarithmic expression and determined the possible original expressions that could result in the given expression. We found that option B, log⁡123\log _{\frac{1}{2}} 3, is the correct original expression. This demonstrates the power of the change of base formula in manipulating logarithmic expressions and finding the original expression that results in a given expression.

References

Further Reading

Introduction

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 provide a Q&A guide to help you understand the change of base formula and its applications.

Q: What is the change of base formula?

A: The change of base formula is a mathematical formula that allows us to express a logarithmic expression in terms of a different base. It is given by:

log⁡ax=log⁡bxlog⁡ba\log_a x = \frac{\log_b x}{\log_b a}

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

A: To apply the change of base formula, you need to follow these steps:

  1. Identify the base and the argument of the logarithm.
  2. Choose a new base for the logarithm.
  3. Apply the change of base formula using the new base.

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

A: The change of base formula has many applications in mathematics, including:

  • Converting between different bases: The change of base formula allows us to convert a logarithmic expression from one base to another.
  • Simplifying logarithmic expressions: The change of base formula can be used to simplify complex logarithmic expressions.
  • Finding the original expression: The change of base formula can be used to find the original expression that results in a given expression.

Q: How do I choose the new base for the logarithm?

A: When choosing the new base for the logarithm, you should consider the following factors:

  • Ease of calculation: Choose a base that is easy to calculate with.
  • Simplification of the expression: Choose a base that simplifies the expression.
  • Convenience: Choose a base that is convenient to work with.

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

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

  • Incorrect application of the formula: Make sure to apply the formula correctly.
  • Failure to simplify the expression: Make sure to simplify the expression after applying the formula.
  • Choosing an inconvenient base: Choose a base that is convenient to work with.

Q: Can the change of base formula be used to find the original expression?

A: Yes, the change of base formula can be used to find the original expression that results in a given expression. To do this, you need to apply the change of base formula in reverse.

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

A: To apply the change of base formula in reverse, you need to follow these steps:

  1. Identify the given expression.
  2. Apply the change of base formula to the given expression.
  3. Simplify the expression.

Q: What are some real-world applications of the change of base formula?

A: The change of base formula has many real-world applications, including:

  • Computer science: The change of base formula is used in computer science to convert between different bases.
  • Engineering: The change of base formula is used in engineering to simplify complex logarithmic expressions.
  • Finance: The change of base formula is used in finance to calculate interest rates and investment returns.

Conclusion

In this article, we provided a Q&A guide to help you understand the change of base formula and its applications. We hope that this guide has been helpful in clarifying any questions you may have had about the change of base formula.

References

Further Reading