How do you solve 4(2^x) = 5^x?

Nov 18, 2015

I found: $x = 1.5129$

Explanation:

Start by writing it as:
${2}^{2} \cdot {2}^{x} = {5}^{x}$
${2}^{x + 2} = {5}^{x}$
take the $\ln$ of both sides:
$\ln \left({2}^{x + 2}\right) = \ln {5}^{x}$
use the property of logs:
$\log {x}^{a} = a \log x$
to get:
$\left(x + 2\right) \ln 2 = x \ln 5$
$x \ln 2 + 2 \ln 2 - x \ln 5 = 0$
isolate $x$:
$x \left(\ln 2 - \ln 5\right) = - 2 \ln 2$
$x = - \frac{2 \ln 2}{\ln 2 - \ln 5} = - \frac{1.3863}{-} 0.9163 = 1.5129$