Find The Value Of $\log _{\frac{1}{2}} 462$ To Four Decimal Places.A. -8.8517 B. -0.1130 C. 0.1130 D. 8.8517

Understanding the Problem

The problem requires finding the value of the logarithm with a base of 1/2, denoted as $\log _{\frac{1}{2}} 462$. This means we need to find the exponent to which the base 1/2 must be raised to obtain the number 462.

Recalling Logarithmic Properties

To solve this problem, we can use the property of logarithms that states $\log_b a = \frac{\log_c a}{\log_c b}$, where $b$, $a$, and $c$ are positive real numbers and $c \neq 1$. This property allows us to change the base of a logarithm to a more familiar base, such as 10 or $e$.

Applying the Change of Base Formula

Using the change of base formula, we can rewrite the given logarithm as:

$$\log _{\frac{1}{2}} 462 = \frac{\log 462}{\log \frac{1}{2}}$$

Evaluating the Logarithms

To evaluate the logarithms in the numerator and denominator, we can use a calculator or a logarithmic table. Let's assume we have a calculator that can evaluate logarithms to four decimal places.

Calculating the Numerator

The numerator is $\log 462$. Using a calculator, we find that:

$$\log 462 \approx 2.6624$$

Calculating the Denominator

The denominator is $\log \frac{1}{2}$. Using a calculator, we find that:

$$\log \frac{1}{2} = -\log 2 \approx -0.3010$$

Calculating the Final Answer

Now that we have the values of the numerator and denominator, we can calculate the final answer:

$$\log _{\frac{1}{2}} 462 = \frac{\log 462}{\log \frac{1}{2}} \approx \frac{2.6624}{-0.3010} \approx -8.8517$$

Conclusion

Therefore, the value of $\log _{\frac{1}{2}} 462$ to four decimal places is approximately -8.8517.

Answer Choices

Comparing our final answer with the answer choices, we see that:

• A. -8.8517 is the correct answer.
• B. -0.1130 is incorrect.
• C. 0.1130 is incorrect.
• D. 8.8517 is incorrect.

Final Answer

The final answer is $\boxed{-8.8517}$.

Understanding the Problem

The problem of finding the value of a logarithm with a base of 1/2 is a common one in mathematics. In this article, we will delve deeper into the concept of logarithms and provide a step-by-step guide on how to solve this type of problem.

Recalling Logarithmic Properties

To solve the problem of finding the value of a logarithm with a base of 1/2, we need to recall the properties of logarithms. One of the most important properties is the change of base formula, which states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $b$, $a$, and $c$ are positive real numbers and $c \neq 1$.

Q&A: Logarithm with a Base of 1/2

Q: What is the value of $\log _{\frac{1}{2}} 462$?

A: The value of $\log _{\frac{1}{2}} 462$ is approximately -8.8517.

Q: How do I solve the problem of finding the value of a logarithm with a base of 1/2?

A: To solve this problem, you need to use the change of base formula, which states that $\log_b a = \frac{\log_c a}{\log_c b}$, where $b$, $a$, and $c$ are positive real numbers and $c \neq 1$. You can then use a calculator or a logarithmic table to evaluate the logarithms in the numerator and denominator.

Q: What is the change of base formula?

A: The change of base formula is $\log_b a = \frac{\log_c a}{\log_c b}$, where $b$, $a$, and $c$ are positive real numbers and $c \neq 1$.

Q: How do I evaluate the logarithms in the numerator and denominator?

A: You can use a calculator or a logarithmic table to evaluate the logarithms in the numerator and denominator.

Q: What is the final answer to the problem of finding the value of a logarithm with a base of 1/2?

A: The final answer to the problem of finding the value of a logarithm with a base of 1/2 is approximately -8.8517.

Common Mistakes to Avoid

When solving the problem of finding the value of a logarithm with a base of 1/2, there are several common mistakes to avoid. These include:

• Not using the change of base formula: The change of base formula is essential in solving this type of problem. Without it, you will not be able to find the value of the logarithm.
• Not evaluating the logarithms correctly: You need to evaluate the logarithms in the numerator and denominator correctly in order to get the final answer.
• Not using a calculator or logarithmic table: A calculator or logarithmic table is necessary in evaluating the logarithms in the numerator and denominator.

Conclusion

In conclusion, finding the value of a logarithm with a base of 1/2 is a common problem in mathematics. By using the change of base formula and evaluating the logarithms correctly, you can solve this type of problem. Remember to avoid common mistakes such as not using the change of base formula and not evaluating the logarithms correctly.

Final Answer

The final answer to the problem of finding the value of a logarithm with a base of 1/2 is approximately -8.8517.