Solve 20000=2000e^20t

$$20000 = 2000 { e }^{ 20t }$$

Solve for t

$$t=\frac{\ln(10)}{20}\approx 0.115129255$$

Steps Using Definition of Logarithm

Divide both sides by 2000.

$$\frac{20000}{2000}=e^{20t}$$

Divide 20000 by 2000 to get 10.

$$10=e^{20t}$$

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

$$e^{20t}=10$$

Take the logarithm of both sides of the equation.

$$\log(e^{20t})=\log(10)$$

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

$$20t\log(e)=\log(10)$$

Divide both sides by \log(e).

$$20t=\frac{\log(10)}{\log(e)}$$

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

$$20t=\log_{e}\left(10\right)$$

Divide both sides by 20.

$$t=\frac{\ln(10)}{20}$$

Solve for t (complex solution)

$$t=\frac{\pi n_{1}i}{10}+\frac{\ln(10)}{20}$$

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