Question

simplify.
$ln e = square$
$ln e^{2x} = square$
$ln 1 = square$
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 $\ln e^x = x$, when $x = 1$, $\ln e = 1$.

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

Using the logarithm property $\ln a^b = b\ln a$, here $a = e$ and $b = 2x$. So $\ln e^{2x}=2x\ln e$. But we know $\ln e = 1$, so $2x\ln e = 2x\times1 = 2x$.

Step3: Simplify $\ln 1$

Recall that $e^0 = 1$, so $\ln 1=\ln e^0$. Using the property $\ln e^x = x$, when $x = 0$, $\ln 1 = 0$.

Answer:

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