# How do you solve 5(2)^(2x)-4=13?

Jan 27, 2017

Please see the explanation.

#### Explanation:

Add 4 to both sides:

$5 {\left(2\right)}^{2 x} = 17$

Divide both sides by 5:

${\left(2\right)}^{2 x} = \frac{17}{5}$

Use the natural logarithm on both sides:

$\ln \left({\left(2\right)}^{2 x}\right) = \ln \left(\frac{17}{5}\right)$

Use the property $\ln \left({a}^{b}\right) = \left(b\right) \ln \left(a\right)$

$\left(2 x\right) \ln \left(\left(2\right)\right) = \ln \left(\frac{17}{5}\right)$

Divide both sides by 2ln(x):

$x = \ln \frac{\frac{17}{5}}{2 \ln \left(2\right)}$

check:

$5 {\left(2\right)}^{2 \left(\ln \frac{\frac{17}{5}}{2 \ln \left(2\right)}\right)} - 4 = 13$

$5 {\left(2\right)}^{\left(\ln \frac{\frac{17}{5}}{\ln \left(2\right)}\right)} - 4 = 13$

5(2)^((log_2(17/5)) - 4 = 13

$5 \left(\frac{17}{5}\right) - 4 = 13$

$17 - 4 = 13$

$13 = 13$

$x = \ln \frac{\frac{17}{5}}{2 \ln \left(2\right)}$ checks.