Certainly! Let's solve [tex]\(\log_{\frac{1}{2}} \sqrt[3]{4}\)[/tex] step by step.
1. Identify the Components:
- The base of the logarithm is [tex]\(\frac{1}{2}\)[/tex].
- The argument of the logarithm is [tex]\(\sqrt[3]{4}\)[/tex].
2. Express the Argument Using Exponents:
- Recall that [tex]\(\sqrt[3]{4}\)[/tex] can be written as [tex]\(4^{1/3}\)[/tex].
So, we have:
[tex]\[
\log_{\frac{1}{2}} 4^{1/3}
\][/tex]
3. Apply the Change of Base Formula:
- The change of base formula states that [tex]\(\log_b a = \frac{\log_c a}{\log_c b}\)[/tex], where [tex]\(c\)[/tex] is any positive number. Here, we usually use the natural logarithm (ln) or the common logarithm (log) for convenience.
Let's use the natural logarithm (ln):
[tex]\[
\log_{\frac{1}{2}} 4^{1/3} = \frac{\ln(4^{1/3})}{\ln(\frac{1}{2})}
\][/tex]
4. Simplify the Numerator:
- We can simplify [tex]\(\ln(4^{1/3})\)[/tex] using the property of logarithms that [tex]\(\ln(a^b) = b \ln(a)\)[/tex].
So,
[tex]\[
\ln(4^{1/3}) = \frac{1}{3} \ln(4)
\][/tex]
5. Substitute Back into the Formula:
Now we have:
[tex]\[
\log_{\frac{1}{2}} 4^{1/3} = \frac{\frac{1}{3} \ln(4)}{\ln(\frac{1}{2})}
\][/tex]
6. Solve the Denominator:
Note that [tex]\(\ln(\frac{1}{2}) = -\ln(2)\)[/tex] because [tex]\(\frac{1}{2} = 2^{-1}\)[/tex] and thus [tex]\(\ln(2^{-1}) = -\ln(2)\)[/tex].
So,
[tex]\[
\ln(\frac{1}{2}) = -\ln(2)
\][/tex]
7. Combine the Results:
[tex]\[
\log_{\frac{1}{2}} 4^{1/3} = \frac{\frac{1}{3} \ln(4)}{-\ln(2)}
\][/tex]
8. Simplify Further:
- Since [tex]\(\ln(4) = \ln(2^2) = 2\ln(2)\)[/tex],
We can rewrite the expression as:
[tex]\[
\log_{\frac{1}{2}} 4^{1/3} = \frac{\frac{1}{3} \cdot 2 \ln(2)}{-\ln(2)}
\][/tex]
[tex]\[
= \frac{\frac{2}{3} \ln(2)}{-\ln(2)}
\][/tex]
9. Cancel out [tex]\(\ln(2)\)[/tex]:
[tex]\[
= \frac{2}{3} \left(-1\right)
\][/tex]
10. Final Result:
[tex]\[
\log_{\frac{1}{2}} 4^{1/3} = -\frac{2}{3}
\][/tex]
So, the final answer is:
[tex]\[
\log_{\frac{1}{2}} \sqrt[3]{4} = -\frac{2}{3}
\][/tex]
Given the numerical result provided, this matches closely, but because of precise numerical representation, it appears in decimal form as approximately [tex]\(-0.6666666666666665\)[/tex].
