Question

find the indicated value of the logarithmic function. \\( \ln (e^{2x}) \\) choose the correct value of the logarithmic function \\( \bigcirc \\) a. \\( e^{\ln (2x)} \\) \\( \bigcirc \\) b. \\( 2x \\) \\( \bigcirc \\) c. \\( \ln (2x) \\) \\( \bigcirc \\) d. \\( e^{2x} \\)

Explanation:

Step1: Recall Logarithm Power Rule

The power rule of logarithms states that $\ln(a^b) = b\ln(a)$. Here, the function is $\ln(e^{2x})$, so $a = e$ and $b = 2x$.

Step2: Apply Inverse Property of Log and Exponential

We know that $\ln(e^y)=y$ because $\ln$ (natural log) and $e^x$ are inverse functions. Substituting $y = 2x$, we get $\ln(e^{2x}) = 2x$.

Answer:

$2x$ (which corresponds to option B)