# How do you solve log (x + 9) - log x = 3?

Mar 12, 2016

Use the log property ${\log}_{a} n - {\log}_{a} m = {\log}_{a} \left(\frac{n}{m}\right)$

#### Explanation:

$\log \left(x + 9\right) - \log \left(x\right) = 3$

$\log \left(\frac{x + 9}{x}\right) = 3$

Since nothing is noted in subscript, the log is in base 10.

$\frac{x + 9}{x} = {10}^{3}$

$\frac{x + 9}{x} = 1000$

We can now solve as a simple linear equation by using the property $\frac{a}{b} = \frac{m}{n} \implies a \times n = b \times m$

$x + 9 = 1000 x$

$9 = 999 x$

$\frac{9}{999} = x$

$\frac{1}{111} = x$

Hopefully this helps!