Solve 20000=2000e^z(20)

$$20000 = 2000 { e }^{ z(20) }$$

Solve for z

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

Steps Using Definition of Logarithm

Divide both sides by 2000.

$$\frac{20000}{2000}=e^{z\times 20}$$

Divide 20000 by 2000 to get 10.

$$10=e^{z\times 20}$$

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

$$e^{z\times 20}=10$$

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

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

Take the logarithm of both sides of the equation.

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

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

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

Divide both sides by \log(e).

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

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

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

Divide both sides by 20.

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

Solve for z (complex solution)

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

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