Solve e^2t=3

$$e ^ { 2 t } = 3$$

Solve for t

$$t=\frac{\ln(3)}{2}\approx 0.549306144$$

Steps Using Definition of Logarithm

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

$$e^{2t}=3$$

Take the logarithm of both sides of the equation.

$$\log(e^{2t})=\log(3)$$

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

$$2t\log(e)=\log(3)$$

Divide both sides by \log(e).

$$2t=\frac{\log(3)}{\log(e)}$$

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

$$2t=\log_{e}\left(3\right)$$

Divide both sides by 2.

$$t=\frac{\ln(3)}{2}$$

Solve for t (complex solution)

$$t=\frac{\ln(3)}{2}+i\pi n_{1}$$

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