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

Jul 4, 2016

$x = - \frac{2 \ln \left(5\right)}{\ln \left(2\right) - \ln \left(5\right)} \approx + 3.513$ to 3 decimal places

#### Explanation:

$\textcolor{b r o w n}{\text{Take logs of both sides}}$

$\text{ } \ln \left({2}^{x}\right) = \ln \left({5}^{x - 2}\right)$

$\textcolor{b r o w n}{\text{This is the same as}}$

$\text{ } x \ln \left(2\right) = \left(x - 2\right) \ln \left(5\right)$

$\textcolor{b r o w n}{\text{Divide both sides by } \ln \left(2\right)}$

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

$\textcolor{b r o w n}{\text{Multiply out the bracket}}$

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

$\textcolor{b r o w n}{\text{Subtract "xln(5)/ln(2)" from both sides}}$

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

$\textcolor{b r o w n}{\text{Factor out the } x}$

$\text{ } x \left(1 - \ln \frac{5}{\ln} \left(2\right)\right) = - \frac{2 \ln \left(5\right)}{\ln} \left(2\right)$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{g r e e n}{\text{Note that } \left(1 - \ln \frac{5}{\ln} \left(2\right)\right) = \frac{\ln \left(2\right) - \ln \left(5\right)}{\ln} \left(2\right)}$

'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b r o w n}{\text{Divide both sides by } \frac{\ln \left(2\right) - \ln \left(5\right)}{\ln} \left(2\right)}$

$\text{ } x = - \frac{2 \ln \left(5\right)}{\cancel{\ln \left(2\right)}} \times \frac{\cancel{\ln \left(2\right)}}{\ln \left(2\right) - \ln \left(5\right)}$

$\text{ } x = - \frac{2 \ln \left(5\right)}{\ln \left(2\right) - \ln \left(5\right)} \approx + 3.513$ to 3 decimal places