Which Is Equivalent To Log ⁡ 1 2 462 \log_{\frac{1}{2}} 462 Lo G 2 1 ​ ​ 462 ?A. Log ⁡ 1 2 462 Log ⁡ 462 1 2 \frac{\log_{\frac{1}{2}} 462}{\log_{462} \frac{1}{2}} L O G 462 ​ 2 1 ​ L O G 2 1 ​ ​ 462 ​ B. Log ⁡ 462 Log ⁡ 1 2 \frac{\log 462}{\log \frac{1}{2}} L O G 2 1 ​ L O G 462 ​ C. Log ⁡ 1 2 Log ⁡ 462 \frac{\log \frac{1}{2}}{\log 462} L O G 462 L O G 2 1 ​ ​

Introduction

Logarithms are a fundamental concept in mathematics, and they have numerous applications in various fields, including science, engineering, and finance. In this article, we will explore the concept of logarithmic equivalents and how to find the equivalent of a given logarithmic expression. We will focus on the specific problem of finding the equivalent of $\log_{\frac{1}{2}} 462$.

What are Logarithmic Equivalents?

Logarithmic equivalents refer to the process of converting a logarithmic expression from one base to another. This is useful when we need to work with logarithmic expressions in different bases or when we need to simplify complex logarithmic expressions.

The Change of Base Formula

The change of base formula is a fundamental concept in logarithms that allows us to convert a logarithmic expression from one base to another. The formula is given by:

$$\log_{b} a = \frac{\log_{c} a}{\log_{c} b}$$

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.

Applying the Change of Base Formula

To find the equivalent of $\log_{\frac{1}{2}} 462$, we can use the change of base formula. We can choose any base $c$ that is convenient for us. Let's choose the base $10$ as our new base.

$$\log_{\frac{1}{2}} 462 = \frac{\log_{10} 462}{\log_{10} \frac{1}{2}}$$

Simplifying the Expression

Now that we have applied the change of base formula, we can simplify the expression further. We can use the properties of logarithms to simplify the expression.

$$\frac{\log_{10} 462}{\log_{10} \frac{1}{2}} = \frac{\log_{10} 462}{\log_{10} 1 - \log_{10} 2}$$

Using the property of logarithms that $\log_{a} 1 = 0$, we can simplify the expression further.

$$\frac{\log_{10} 462}{\log_{10} 1 - \log_{10} 2} = \frac{\log_{10} 462}{-\log_{10} 2}$$

Comparing the Options

Now that we have simplified the expression, we can compare it with the given options.

A. $\frac{\log_{\frac{1}{2}} 462}{\log_{462} \frac{1}{2}}$

B. $\frac{\log 462}{\log \frac{1}{2}}$

C. $\frac{\log \frac{1}{2}}{\log 462}$

We can see that option A is not equivalent to our simplified expression. Option B is equivalent to our simplified expression, but it is not in the correct form. Option C is not equivalent to our simplified expression.

Conclusion

In conclusion, the equivalent of $\log_{\frac{1}{2}} 462$ is $\frac{\log 462}{\log \frac{1}{2}}$. This can be verified by applying the change of base formula and simplifying the expression.

Final Answer

The final answer is $\boxed{B}$.

Additional Information

• The change of base formula is a fundamental concept in logarithms that allows us to convert a logarithmic expression from one base to another.
• The change of base formula is given by $\log_{b} a = \frac{\log_{c} a}{\log_{c} b}$, where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.
• The change of base formula can be used to simplify complex logarithmic expressions and to convert logarithmic expressions from one base to another.

References

• [1] "Logarithms" by Khan Academy
• [2] "Change of Base Formula" by Math Is Fun
• [3] "Logarithmic Equivalents" by Wolfram MathWorld
  Logarithmic Equivalents: A Q&A Guide =====================================

Introduction

In our previous article, we explored the concept of logarithmic equivalents and how to find the equivalent of a given logarithmic expression. In this article, we will provide a Q&A guide to help you better understand the concept of logarithmic equivalents.

Q: What is a logarithmic equivalent?

A: A logarithmic equivalent is a process of converting a logarithmic expression from one base to another. This is useful when we need to work with logarithmic expressions in different bases or when we need to simplify complex logarithmic expressions.

Q: What is the change of base formula?

A: The change of base formula is a fundamental concept in logarithms that allows us to convert a logarithmic expression from one base to another. The formula is given by:

$$\log_{b} a = \frac{\log_{c} a}{\log_{c} b}$$

where $a$, $b$, and $c$ are positive real numbers, and $c \neq 1$.

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

A: To apply the change of base formula, you need to choose a new base $c$ that is convenient for you. Then, you can use the formula to convert the logarithmic expression from the original base to the new base.

Q: What are some common bases used in logarithmic equivalents?

A: Some common bases used in logarithmic equivalents include:

• Base 10 (common logarithm)
• Base 2 (binary logarithm)
• Base $e$ (natural logarithm)

Q: How do I simplify a logarithmic expression using the change of base formula?

A: To simplify a logarithmic expression using the change of base formula, you need to follow these steps:

1. Choose a new base $c$ that is convenient for you.
2. Apply the change of base formula to convert the logarithmic expression from the original base to the new base.
3. Simplify the expression using the properties of logarithms.

Q: What are some common properties of logarithms that I can use to simplify logarithmic expressions?

A: Some common properties of logarithms that you can use to simplify logarithmic expressions include:

• $\log_{a} 1 = 0$
• $\log_{a} a = 1$
• $\log_{a} (bc) = \log_{a} b + \log_{a} c$
• $\log_{a} \left(\frac{b}{c}\right) = \log_{a} b - \log_{a} c$

Q: How do I choose the correct base for a logarithmic expression?

A: To choose the correct base for a logarithmic expression, you need to consider the following factors:

• The original base of the logarithmic expression
• The new base that you want to convert to
• The properties of logarithms that you want to use to simplify the expression

Q: What are some real-world applications of logarithmic equivalents?

A: Logarithmic equivalents have numerous real-world applications, including:

• Finance: Logarithmic equivalents are used to calculate interest rates and investment returns.
• Science: Logarithmic equivalents are used to calculate the pH of a solution and the concentration of a substance.
• Engineering: Logarithmic equivalents are used to calculate the gain of an amplifier and the frequency response of a system.

Conclusion

In conclusion, logarithmic equivalents are a powerful tool for simplifying complex logarithmic expressions and converting logarithmic expressions from one base to another. By understanding the change of base formula and the properties of logarithms, you can apply logarithmic equivalents to a wide range of real-world problems.

Final Tips

• Practice, practice, practice: The more you practice applying logarithmic equivalents, the more comfortable you will become with the concept.
• Use online resources: There are many online resources available that can help you learn logarithmic equivalents, including video tutorials and practice problems.
• Seek help when needed: If you are struggling with logarithmic equivalents, don't be afraid to seek help from a teacher or tutor.