To express [tex]\( \ln \sqrt[6]{5} \)[/tex] as a product and simplify the expression, follow these steps:
1. Rewrite the expression inside the natural logarithm:
The sixth root of [tex]\( 5 \)[/tex] can be written as [tex]\( 5^{1/6} \)[/tex]. Therefore, we have:
[tex]\[
\ln \sqrt[6]{5} = \ln (5^{1/6})
\][/tex]
2. Use the logarithm power rule:
The power rule for logarithms states that [tex]\( \ln (a^b) = b \cdot \ln (a) \)[/tex]. Applying this rule, we get:
[tex]\[
\ln (5^{1/6}) = \frac{1}{6} \cdot \ln 5
\][/tex]
So, the expression [tex]\( \ln \sqrt[6]{5} \)[/tex] can be simplified and expressed as:
[tex]\[
\ln \sqrt[6]{5} = \frac{1}{6} \cdot \ln 5
\][/tex]
This is the final simplified form.
