Solve 4=e^7r

$$4 = e ^ { 7 r }$$

Solve for r

$$r=\frac{2\ln(2)}{7}\approx 0.198042052$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$e^{7r}=4$$

Take the logarithm of both sides of the equation.

$$\log(e^{7r})=\log(4)$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$7r\log(e)=\log(4)$$

Divide both sides by \log(e).

$$7r=\frac{\log(4)}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$7r=\log_{e}\left(4\right)$$

Divide both sides by 7.

$$r=\frac{2\ln(2)}{7}$$

Solve for r (complex solution)

$$r=\frac{i\times 2\pi n_{1}}{7}+\frac{2\ln(2)}{7}$$

$$n_{1}\in \mathrm{Z}$$