Solve 8000=5000(1.005){(e)^12x}

$$8000=5000(1.005) {(e)^{ 12x }}$$

Solve for x

$$x=\frac{\ln(\frac{320}{201})}{12}\approx 0.038751341$$

Steps Using Definition of Logarithm

Multiply 5000 and 1.005 to get 5025.

$$8000=5025e^{12x}$$

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

$$5025e^{12x}=8000$$

Divide both sides by 5025.

$$e^{12x}=\frac{8000}{5025}$$

Reduce the fraction \frac{8000}{5025} to lowest terms by extracting and canceling out 25.

$$e^{12x}=\frac{320}{201}$$

Take the logarithm of both sides of the equation.

$$\log(e^{12x})=\log(\frac{320}{201})$$

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

$$12x\log(e)=\log(\frac{320}{201})$$

Divide both sides by \log(e).

$$12x=\frac{\log(\frac{320}{201})}{\log(e)}$$

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

$$12x=\log_{e}\left(\frac{320}{201}\right)$$

Divide both sides by 12.

$$x=\frac{\ln(\frac{320}{201})}{12}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{6}+\frac{\ln(\frac{320}{201})}{12}$$

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