Solve 3e^5x=1977

$$3 { e }^{ 5x } =1977$$

Solve for x

$$x=\frac{\ln(659)}{5}\approx 1.298144707$$

Steps Using Definition of Logarithm

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

$$3e^{5x}=1977$$

Divide both sides by 3.

$$e^{5x}=659$$

Take the logarithm of both sides of the equation.

$$\log(e^{5x})=\log(659)$$

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

$$5x\log(e)=\log(659)$$

Divide both sides by \log(e).

$$5x=\frac{\log(659)}{\log(e)}$$

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

$$5x=\log_{e}\left(659\right)$$

Divide both sides by 5.

$$x=\frac{\ln(659)}{5}$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{5}+\frac{\ln(659)}{5}$$

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