# How do you solve ln x = -5?

Dec 19, 2015

$x = \frac{1}{e} ^ 5$

#### Explanation:

From the definition of a logarithm, we have the property ${e}^{\ln} \left(x\right) = x$. From this:

$\ln \left(x\right) = - 5$

$\implies {e}^{\ln \left(x\right)} = {e}^{- 5}$

$\implies x = \frac{1}{e} ^ 5$

Dec 19, 2015

$x = {e}^{- 5} \cong 0.6738 \times {10}^{- 2}$

#### Explanation:

If
$\textcolor{w h i t e}{\text{XXX}} \ln \left(x\right) = - 5$
then
$\textcolor{w h i t e}{\text{XXX}} {e}^{\ln \left(x\right)} = {e}^{- 5}$
which can be evaluated using a calculator as $0.006738$

Why?

1. $\ln \left(x\right)$ means the same thing as ${\log}_{e} \left(x\right)$
2. ${b}^{{\log}_{b} \left(a\right)} = a$
$\textcolor{w h i t e}{\text{XXX}}$because ${\log}_{b} \left(a\right)$ means
$\textcolor{w h i t e}{\text{XXX}}$the value, $c$, needed as an exponent to make ${b}^{c} = a$