To find the value of [tex]\(\log _{\frac{1}{2}} 462\)[/tex] to four decimal places, follow these steps:
1. Understand the Logarithm Definition: The expression [tex]\(\log _{\frac{1}{2}} 462\)[/tex] asks for the exponent to which the base [tex]\(\frac{1}{2}\)[/tex] must be raised to yield 462.
2. Apply the Change of Base Formula: We can use the change of base formula to evaluate logarithms with bases other than 10 or [tex]\(e\)[/tex]. The change of base formula is:
[tex]\[
\log_b a = \frac{\log_c a}{\log_c b}
\][/tex]
where [tex]\(c\)[/tex] can be any positive number, commonly 10 (common logarithm) or [tex]\(e\)[/tex] (natural logarithm).
3. Choose Logarithm Base 10 for Calculation:
[tex]\[
\log_{\frac{1}{2}} 462 = \frac{\log_{10} 462}{\log_{10} \frac{1}{2}}
\][/tex]
4. Evaluate the Logarithms:
- [tex]\(\log_{10} 462\)[/tex] is straightforward to compute using a calculator or logarithm table.
- [tex]\(\log_{10} \frac{1}{2}\)[/tex] also needs to be computed.
5. Calculate the Intermediate Values:
- [tex]\(\log_{10} 462\)[/tex] might yield a positive number since 462 is greater than 1.
- [tex]\(\log_{10} \frac{1}{2}\)[/tex] will yield a negative number since [tex]\(\frac{1}{2}\)[/tex] is a fraction less than 1, specifically:
[tex]\[
\log_{10} \frac{1}{2} = \log_{10} 1 - \log_{10} 2 = 0 - \log_{10} 2 = -\log_{10} 2
\][/tex]
6. Combine the Results:
[tex]\[
\log_{\frac{1}{2}} 462 = \frac{\log_{10} 462}{-\log_{10} 2}
\][/tex]
7. Perform the Division and Simplify:
Using a calculator, you would get a precise value after calculation. However, for simplicity, I'll present it directly.
[tex]\[
\log_{\frac{1}{2}} 462 \approx -8.8517
\][/tex]
Thus, the value of [tex]\(\log _{\frac{1}{2}} 462\)[/tex] to four decimal places is [tex]\(-8.8517\)[/tex]. Therefore, the correct answer is [tex]\(-8.8517\)[/tex].
