Question

simplify
\ln e =
\ln e^{2x} =
\ln 1 =
done

Explanation:

Step1: Use natural log base rule

Recall that $\ln e = \log_e e$, and by definition $\log_b b = 1$. So $\ln e = 1$.

Step2: Use log power & base rule

Apply the logarithm power rule $\ln a^b = b\ln a$, then use $\ln e =1$:
$\ln e^{2x} = 2x \cdot \ln e = 2x \cdot 1 = 2x$

Step3: Use natural log of 1 rule

By definition, $\ln 1 = \log_e 1$, and any log of 1 is 0. So $\ln 1 = 0$.

Answer:

$\ln e = 1$
$\ln e^{2x} = 2x$
$\ln 1 = 0$