To simplify the expression [tex]\(\frac{\log \sqrt[3]{8}}{\log 4}\)[/tex], let's break it down step by step.
1. Numerator Analysis:
- We start by analyzing [tex]\(\log \sqrt[3]{8}\)[/tex]. First, note that [tex]\(\sqrt[3]{8}\)[/tex] represents the cube root of 8.
- Since [tex]\(8 = 2^3\)[/tex], the cube root of 8 can be expressed as:
[tex]\[
\sqrt[3]{8} = (2^3)^{1/3} = 2
\][/tex]
- Thus, [tex]\(\log \sqrt[3]{8} = \log 2\)[/tex].
2. Denominator Analysis:
- Now, let's consider [tex]\(\log 4\)[/tex]. Similarly, we note that [tex]\(4 = 2^2\)[/tex].
- Using the logarithm property that [tex]\(\log a^b = b \log a\)[/tex], we have:
[tex]\[
\log 4 = \log (2^2) = 2 \log 2
\][/tex]
3. Putting It All Together:
- We now substitute these simplified forms back into our original expression:
[tex]\[
\frac{\log \sqrt[3]{8}}{\log 4} = \frac{\log 2}{2 \log 2}
\][/tex]
- The [tex]\(\log 2\)[/tex] terms in the numerator and the denominator cancel out, leaving:
[tex]\[
\frac{\log 2}{2 \log 2} = \frac{1}{2}
\][/tex]
Therefore, the simplified form of [tex]\(\frac{\log \sqrt[3]{8}}{\log 4}\)[/tex] is:
[tex]\[
\frac{1}{2}
\][/tex]
The numerical values involved in following through these steps are:
- The numerator [tex]\(\log 2\)[/tex] is approximately [tex]\(0.6931\)[/tex].
- The denominator [tex]\(2 \log 2\)[/tex] is approximately [tex]\(1.3863\)[/tex].
Thus, our final simplified value accurately turns out to be [tex]\(0.5\)[/tex].
