# How do you solve log_7(x-3)-log_7x=3?

Dec 1, 2016

$x = - \frac{1}{114}$

#### Explanation:

Condense the logs on the left -- Subtracting logs of the same base can be rewritten as dividing within the log:
${\log}_{a} b - {\log}_{a} c = {\log}_{a} \left(\frac{b}{c}\right)$

${\log}_{7} \left(x - 3\right) - {\log}_{7} x = 3$
${\log}_{7} \left(\frac{x - 3}{x}\right) = 3$

Now use the log rule: If ${\log}_{a} b = n$, then ${a}^{n} = b$.
${7}^{3} = \frac{x - 3}{x}$

$343 = \frac{x - 3}{x}$

Multiply each side by x:
$343 x = x - 3$

Subtract x from each side:
$342 x = - 3$

$x = - \frac{3}{342}$

$x = - \frac{1}{114}$