# How do you simplify 4lne+lne?

Mar 30, 2016

$4 \ln e + \ln e = \textcolor{g r e e n}{5}$

#### Explanation:

$\ln \textcolor{red}{e}$
$\textcolor{w h i t e}{\text{XXX}} = {\log}_{\textcolor{b l u e}{e}} \textcolor{red}{e}$

Remembering that
$\textcolor{w h i t e}{\text{XXX}} {\log}_{\textcolor{b l u e}{b}} \textcolor{red}{a} = c$
means
$\textcolor{w h i t e}{\text{XXX}} {\textcolor{b l u e}{b}}^{c} = \textcolor{red}{a}$

Therefore
$\textcolor{w h i t e}{\text{XXX}} {\log}_{\textcolor{b l u e}{e}} \textcolor{red}{e} = c$
means
$\textcolor{w h i t e}{\text{XXX}} {\textcolor{b l u e}{e}}^{c} = \textcolor{red}{e}$

That is
$\textcolor{w h i t e}{\text{XXX}} \textcolor{g r e e n}{{\ln}_{e}} = {\log}_{e} e \left(= c\right) = \textcolor{g r e e n}{1}$

Therefore
$\textcolor{w h i t e}{\text{XXX}} 4 \textcolor{g r e e n}{{\ln}_{e}} + \textcolor{g r e e n}{{\ln}_{e}}$

$\textcolor{w h i t e}{\text{XXX}} = 4 \left(\textcolor{g r e e n}{1}\right) + \left(\textcolor{g r e e n}{1}\right)$

$\textcolor{w h i t e}{\text{XXX}} = 5$