# How do you solve 4^ { 8x } = 500?

Oct 19, 2017

One can use any base logarithm that one likes on both sides:

${\log}_{b} \left({4}^{8 x}\right) = {\log}_{b} \left(500\right)$

Use the property ${\log}_{b} \left({A}^{C}\right) = \left(C\right) {\log}_{b} \left(A\right)$

$\left(8 x\right) {\log}_{b} \left(4\right) = {\log}_{b} \left(500\right)$

Divide both sides by $8 {\log}_{b} \left(4\right)$:

$x = {\log}_{b} \frac{500}{8 {\log}_{b} \left(4\right)}$

Most calculators have base 10 and base e, therefore, I recommend that you use one of these two bases, for an exact representation of x:

$x = {\log}_{10} \frac{500}{8 {\log}_{10} \left(4\right)} = \ln \frac{500}{8 \ln \left(4\right)}$

Here is an approximate number for x:

$x \approx 0.560362$

Oct 19, 2017

$x = 0.56036$

#### Explanation:

${4}^{8 x} = 500$

Taking log on both sides,
$\log {4}^{8 x} = \log 500$

$8 x \log 4 = \log 500$

$8 x = \log \frac{500}{\log} 4 = \frac{2.699}{0.6021} = 4.4829$

$x = \frac{4.4829}{8} \approx 0.56036$