# How do you solve ln(x+1)-lnx=5?

Jun 9, 2017

See below.

#### Explanation:

When we subtract two natural logs (or two logs in general), we divide the expressions on the inside.

$\ln \left(x + 1\right) - \ln x = 5$

$\ln \left(\frac{x + 1}{x}\right) = 5$

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

$x {e}^{5} = x + 1$

$x {e}^{5} - x = 1$

$x \left({e}^{5} - 1\right) = 1$

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

Jun 9, 2017

I got: $x = \frac{1}{{e}^{5} - 1} = 0.006783$

#### Explanation:

We can use a property of logs:

$\log x - \log y = \log \left(\frac{x}{y}\right)$

and write:

$\ln \left(\frac{x + 1}{x}\right) = 5$

use the definition of (natural) log:

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

rearrange:

$x + 1 = x {e}^{5}$

$x \left({e}^{5} - 1\right) = 1$

$x = \frac{1}{{e}^{5} - 1} = 0.006783$