Solve 8000=5000(1005){(e)^12x}

$$8000=5000(1005) {(e)^{ 12x }}$$

Solve for x

$$x=\frac{\ln(2)}{4}-\frac{\ln(5025)}{12}\approx -0.536894933$$

Steps Using Definition of Logarithm

Multiply 5000 and 1005 to get 5025000.

$$8000=5025000e^{12x}$$

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

$$5025000e^{12x}=8000$$

Divide both sides by 5025000.

$$e^{12x}=\frac{8000}{5025000}$$

Reduce the fraction \frac{8000}{5025000} to lowest terms by extracting and canceling out 1000.

$$e^{12x}=\frac{8}{5025}$$

Take the logarithm of both sides of the equation.

$$\log(e^{12x})=\log(\frac{8}{5025})$$

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

$$12x\log(e)=\log(\frac{8}{5025})$$

Divide both sides by \log(e).

$$12x=\frac{\log(\frac{8}{5025})}{\log(e)}$$

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

$$12x=\log_{e}\left(\frac{8}{5025}\right)$$

Divide both sides by 12.

$$x=\frac{\ln(\frac{8}{5025})}{12}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{6}+\frac{\ln(2)}{4}-\frac{\ln(5025)}{12}$$

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