# How do you solve 3^(2-x)=5^(2x+1)?

Dec 19, 2015

I found $x = 0.13614$

#### Explanation:

We can try taking the natural log of both sides:
$\ln \left({3}^{2 - x}\right) = \ln \left({5}^{2 x + 1}\right)$
then use the property of logs:
$\log {x}^{y} = y \log x$ to write:
$\left(2 - x\right) \ln \left(3\right) = \left(2 x + 1\right) \ln \left(5\right)$
$2 \ln \left(3\right) - x \ln \left(3\right) = 2 x \ln \left(5\right) + \ln \left(5\right)$
collecting $x$:
$x \left(2 \ln \left(5\right) + \ln \left(3\right)\right) = 2 \ln \left(3\right) - \ln \left(5\right)$
so:
$x = \frac{2 \ln \left(3\right) - \ln \left(5\right)}{2 \ln \left(5\right) + \ln \left(3\right)} = 0.13614$