Sure, let's evaluate the function [tex]\( g(x) = \frac{\ln \sqrt[3]{x^2}}{x^4} \)[/tex] at [tex]\( x = 2 \)[/tex].
To do this, let's break it down into manageable steps:
1. Simplify the expression inside the logarithm:
[tex]\[
\sqrt[3]{x^2} \text{ can be written as } (x^2)^{1/3}
\][/tex]
Which is:
[tex]\[
(x^2)^{1/3} = x^{2/3}
\][/tex]
2. Apply the logarithm to the simplified expression:
[tex]\[
\ln(x^{2/3}) \text{ can be simplified using logarithm properties as } \frac{2}{3} \ln(x)
\][/tex]
3. Substitute this back into the original function:
[tex]\[
g(x) = \frac{\frac{2}{3} \ln(x)}{x^4}
\][/tex]
4. Evaluate [tex]\( g(x) \)[/tex] at [tex]\( x = 2 \)[/tex]:
[tex]\[
g(2) = \frac{\frac{2}{3} \ln(2)}{2^4}
\][/tex]
5. Simply compute the values:
- Calculate [tex]\( 2^4 \)[/tex]:
[tex]\[
2^4 = 16
\][/tex]
- Let’s find [tex]\( \ln(2) \)[/tex]. The natural logarithm of 2 ([tex]\( \ln(2) \)[/tex]) is approximately [tex]\( 0.693147 \)[/tex].
- Then, compute [tex]\( \frac{2}{3} \ln(2) \)[/tex]:
[tex]\[
\frac{2}{3} \times 0.693147 = 0.462098
\][/tex]
6. Substitute all these values back into the equation:
[tex]\[
g(2) = \frac{0.462098}{16} \approx 0.028881132523331052
\][/tex]
Hence, the evaluated value of the function [tex]\( g(x) \)[/tex] at [tex]\( x = 2 \)[/tex] is:
[tex]\[
g(2) \approx 0.028881132523331052
\][/tex]
