# How do you simplify Ln e^3?

Aug 4, 2016

$\ln \left({e}^{3}\right) = 3$

#### Explanation:

By definition, ${\log}_{a} \left(x\right)$ is the value such that ${a}^{{\log}_{a} \left(x\right)} = x$
From this, it should be clear that for any valid $a$ and $b$, ${\log}_{a} \left({a}^{b}\right) = b$, as ${\log}_{a} \left({a}^{b}\right)$ is the value such that ${a}^{{\log}_{a} \left({a}^{b}\right)} = {a}^{b}$.

As the natural logarithm $\ln$ is just another way of writing the base-$e$ logarithm ${\log}_{e}$, we have

$\ln \left({e}^{3}\right) = {\log}_{e} \left({e}^{3}\right) = 3$