Solve e^5x=7

$$e ^ { 5 x } = 7$$

Solve for x

$$x=\frac{\ln(7)}{5}\approx 0.38918203$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$e^{5x}=7$$

Take the logarithm of both sides of the equation.

$$\log(e^{5x})=\log(7)$$

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

$$5x\log(e)=\log(7)$$

Divide both sides by \log(e).

$$5x=\frac{\log(7)}{\log(e)}$$

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

$$5x=\log_{e}\left(7\right)$$

Divide both sides by 5.

$$x=\frac{\ln(7)}{5}$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{5}+\frac{\ln(7)}{5}$$

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