How do you solve for x in  ln(x+2) - ln(x-2) = ln3?

Apr 16, 2016

$x = 4$

Explanation:

One $\log$ minus another $\log$ can be condensed as the inside of the first divided by the inside of the second, all inside a single $\log$, so

$\ln \left(x + 2\right) - \ln \left(x - 2\right) = \ln \left(\frac{x + 2}{x - 2}\right)$

which gives us

$\ln \left(\frac{x + 2}{x - 2}\right) = \ln 3$.

Raising both sides by $e$ and using simple algebraic manipulation, we can solve for $x$,

$\frac{x + 2}{x - 2} = 3$

$x + 2 = 3 \left(x - 2\right)$
$x + 2 = 3 x - 6$
$8 = 2 x$
$4 = x$