# How do you solve 7=5e^(0.2x)?

Dec 13, 2015

$x = 5 \ln \left(\frac{7}{5}\right) = \ln \left(\frac{16807}{3125}\right)$

#### Explanation:

$\left[1\right] \text{ } 7 = 5 {e}^{0.2 x}$

Divide both sides by 5.

$\left[2\right] \text{ } \frac{7}{5} = \frac{\cancel{5} {e}^{0.2 x}}{\cancel{5}}$

$\left[3\right] \text{ } \frac{7}{5} = {e}^{0.2 x}$

Change to log form.

$\left[4\right] \text{ } \Leftrightarrow {\log}_{e} \left(\frac{7}{5}\right) = 0.2 x$

$\left[5\right] \text{ } \ln \left(\frac{7}{5}\right) = 0.2 x$

Divide both sides by 0.2.

$\left[6\right] \text{ } \ln \frac{\frac{7}{5}}{0.2} = \frac{\cancel{0.2} x}{\cancel{0.2}}$

$\left[7\right] \text{ } x = \ln \frac{\frac{7}{5}}{0.2}$

Simplify.

$\left[8\right] \text{ } x = \ln \frac{\frac{7}{5}}{\frac{1}{5}}$

$\left[9\right] \text{ } \textcolor{b l u e}{x = 5 \ln \left(\frac{7}{5}\right)}$

This is optional, but you can continue by doing this.

Property of log: $n {\log}_{b} a = {\log}_{b} {a}^{n}$

$\left[10\right] \text{ } x = \ln {\left(\frac{7}{5}\right)}^{5}$

$\left[11\right] \text{ } \textcolor{b l u e}{x = \ln \left(\frac{16807}{3125}\right)}$