# How do you solve 6.5^x = 8.4^(4x-10)?

Mar 4, 2016

I found $x = 3.2$

#### Explanation:

Apply the natural log to both sides:
$\ln {\left(6.5\right)}^{x} = \ln {\left(8.4\right)}^{4 x - 10}$
Use the fact that:
$\log {\left(a\right)}^{x} = x \log \left(a\right)$
So we get:
$x \ln \left(6.5\right) = \left(4 x - 10\right) \ln \left(8.4\right)$
Rearrange:
$x \ln \left(6.5\right) - 4 x \ln \left(8.4\right) = - 10 \ln \left(8.4\right)$
$x \left[\ln \left(6.5\right) - 4 \ln \left(8.4\right)\right] = - 10 \ln \left(8.4\right)$
$x = \frac{- 10 \ln \left(8.4\right)}{\ln \left(6.5\right) - 4 \ln \left(8.4\right)} = 3.2$