# How do you solve 5^(x-2) = 3^(3x+2)?

Dec 31, 2015

I found $- 3.2116$

#### Explanation:

We can take the natural log of both sides:
$\ln {\left(5\right)}^{x - 2} = \ln {\left(3\right)}^{3 x + 2}$
We use the property of the logs:
$\log {x}^{y} = y \log x$
$\left(x - 2\right) \ln \left(5\right) = \left(3 x + 2\right) \ln \left(3\right)$
Rearrange:
$x \ln 5 - 2 \ln 5 = 3 x \ln 3 + 2 \ln 3$
$x \left(\ln 5 - 3 \ln 3\right) = 2 \ln 5 + 2 \ln 3$
$x = \frac{2 \ln 5 + 2 \ln 3}{\ln 5 - 3 \ln 3} = - 3.2116$