# How do you solve for x in 2^x = 5^(x+6)?

May 20, 2018

Taking the logarithm on both sides we get
$x \ln \left(2\right) = \left(x + 6\right) \ln \left(5\right)$
rearranging we obtain
$x \left(\ln \left(2\right) - \ln \left(5\right)\right) = 6 \ln \left(5\right)$
so
$x = \frac{6 \ln \left(5\right)}{\ln \left(2\right) - \ln \left(5\right)}$

#### Explanation:

We used that $\ln \left({x}^{r}\right) = r \ln \left(x\right)$