Solve 20000=2000e^10t

$$20000 = 2000 { e }^{ 10t }$$

Solve for t

$$t=\frac{\ln(10)}{10}\approx 0.230258509$$

Steps Using Definition of Logarithm

Divide both sides by 2000.

$$\frac{20000}{2000}=e^{10t}$$

Divide 20000 by 2000 to get 10.

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

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

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

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(e).

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

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

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

Divide both sides by 10.

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

Solve for t (complex solution)

$$t=\frac{\pi n_{1}i}{5}+\frac{\ln(10)}{10}$$

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