# How do you solve ln(x-6) - ln(5) = ln(7) + ln(x-2)?

Jul 10, 2016

$x = \frac{65}{34}$

#### Explanation:

Subtraction of logs is the consequence of the source values being divided. So $\ln \left(x - 6\right) - \ln \left(5\right) \to \ln \left(\frac{x - 5}{5}\right)$

Addition of logs is the consequence of the source values being multiplied. So $\ln \left(7\right) + \ln \left(x - 2\right) \to \ln \left(7 \left(x - 2\right)\right)$

Putting this all together

$\ln \left(\frac{x - 5}{5}\right) = \ln \left(7 \left(x - 2\right)\right)$

$\implies \frac{x - 5}{5} = 7 \left(x - 2\right)$

$x - 5 = 35 \left(x - 2\right)$

$x - 5 = 35 x - 70$

$34 x = 65$

$x = \frac{65}{34}$