# How do you solve ln 3x = ln 15?

Oct 29, 2015

$x = 5$

#### Explanation:

In general if for some function
$\textcolor{w h i t e}{\text{XXX}} f \left(a\right) = \left(b\right)$
then one solution for $a$ is $a = b$
and if for this same function $f \left(c\right)$ is unique for $\forall c$
then $a = b$ is a unique solution.

$\ln \left(a\right)$ is a function of this form.

So
$\textcolor{w h i t e}{\text{XXX}} \ln \left(3 x\right) = \ln \left(15\right)$
implies
$\textcolor{w h i t e}{\text{XXX}} 3 x = 15$

Dividing both sides by 3 gives
$\textcolor{w h i t e}{\text{XXX}} x = 5$