# How do you solve log(x + 10) - log(x) = 2log(5)?

Sep 11, 2015

$x = \frac{5}{12}$

#### Explanation:

$\log \left(x + 10\right) - \log x = 2 \log 5$

Our first step is to rewrite the equation using some laws of logarithms, specifically, $\log A - \log B = \log \left(\frac{A}{B}\right)$. This law allows us to rewrite the left-hand side of the equation as

$\log \left(\frac{x + 10}{x}\right) = 2 \log 5$.

Another law of logarithms, $A \log B = \log {B}^{A}$, allows us to rewrite the right-hand side equivalently as

$\log \left(\frac{x + 10}{x}\right) = \log {5}^{2}$
$= \log 25$

Now, you didn't specify a base for the $\log$ function here, so I will assume that $\log$ means base-2 logarithm. Still, whether the base is 2, 10, e, or whatever, it actually doesn't matter... the answer will be the same. You'll see why in a moment.

Raise 2 to both sides:

${2}^{\log} \left(\frac{x + 10}{x}\right) = {2}^{\log} 25$

The exponential and the logarithm are inverse functions, so the base-2 and the logarithms will cancel:

$\frac{x + 10}{x} = 25$

From here, we just need to use some simple algebra, multiplying both sides by $x$:

$x + 10 = 25 x$

and then subtracting $x$ from both sides:

$10 = 24 x$

And then simplify to arrive at our final answer $x$:

$x = \frac{5}{12}$