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

Dec 2, 2015

$6 {\log}_{\frac{2}{5}} 5$

#### Explanation:

${2}^{x} = {5}^{x + 6}$
$\implies {\log}_{5} {2}^{x} = x + 6$ (logarithm definition)
$x {\log}_{5} 2 = x + 6$ (the logarithm of the power of a number)
Divide both sides into $x$:
$\implies \frac{1}{x} \cdot \left(x {\log}_{5} 2\right) = \frac{1}{x} \cdot \left(x + 6\right)$
$\implies \frac{x + 6}{x} = {\log}_{5} 2$
Add $- 1$ to both sides:
$\frac{x + 6}{x} - 1 = {\log}_{5} 2 - 1$
$\implies \frac{6}{x} = {\log}_{5} 2 - 1$
$\implies x = \frac{6}{{\log}_{5} 2 - 1}$
$= \frac{6}{{\log}_{5} 2 - {\log}_{5} 5}$
$= \frac{6}{\log} _ 5 \left(\frac{2}{5}\right)$
$= 6 {\log}_{\frac{2}{5}} 5$