Question

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

Explanation:

Step1: Simplify $\ln e$

Recall the property of logarithms: $\ln a^b = b\ln a$ and $\ln e = 1$ (since $\ln$ is the natural logarithm with base $e$, and $\log_b b = 1$). For $\ln e$, we can think of it as $\ln e^1$, so using the property $\ln a^b = b\ln a$, we get $1\times\ln e$. But we know $\ln e = 1$, so $\ln e = 1$.

Step2: Simplify $\ln e^{2x}$

Using the logarithm property $\ln a^b = b\ln a$, here $a = e$ and $b = 2x$. So $\ln e^{2x}=2x\times\ln e$. Since $\ln e = 1$, this simplifies to $2x$.

Step3: Simplify $\ln 1$

We know that for any base $b>0, b
eq1$, $\log_b 1 = 0$ because $b^0 = 1$. For the natural logarithm (base $e$), $\ln 1 = 0$ because $e^0 = 1$.

Answer:

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