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

Nov 28, 2015

I found: $x = 0.61454$

#### Explanation:

Take the natural log of both sides:
$\ln {3}^{2 x - 1} = \ln {5}^{2 - 3 x}$
use the property of logs:
$\log {x}^{a} = a \log x$
$\left(2 x - 1\right) \ln 3 = \left(2 - 3 x\right) \ln 5$
$2 \left(\ln 3\right) x - \ln 3 = 2 \ln 5 - 3 \left(\ln 5\right) x$
collect $x$:
$x \left[2 \left(\ln 3\right) + 3 \left(\ln 5\right)\right] = 2 \ln 5 + \ln 3$
$x \left[7.02554\right] = 4.31748$
$x = \frac{4.31748}{7.02554} = 0.61454$