# How do you solve ln(x-3)-ln(x-5)=ln5?

Jul 22, 2016

$x = \frac{11}{2}$

#### Explanation:

The solution must be:

$x - 3 > 0 \mathmr{and} x - 5 > 0$

that's $x > 5$

Since $\log a - \log b = \log \left(\frac{a}{b}\right)$

the given equation is equivalent to:

$\ln \left(\frac{x - 3}{x - 5}\right) = \ln 5$

$\frac{x - 3}{x - 5} = 5$

$x - 3 = 5 \left(x - 5\right)$

$x - 3 = 5 x - 25$

$4 x = 22$

$x = \frac{11}{2}$

that's >5