Solve 5=2e^0.08x

$$5=2 { e }^{ 0.08x }$$

Solve for x

$$x=\frac{25\ln(5)-25\ln(2)}{2}\approx 11.453634148$$

Steps Using Definition of Logarithm

Divide both sides by 2.

$$\frac{5}{2}=e^{0.08x}$$

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

$$e^{0.08x}=\frac{5}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.08x})=\log(\frac{5}{2})$$

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

$$0.08x\log(e)=\log(\frac{5}{2})$$

Divide both sides by \log(e).

$$0.08x=\frac{\log(\frac{5}{2})}{\log(e)}$$

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

$$0.08x=\log_{e}\left(\frac{5}{2}\right)$$

Divide both sides of the equation by 0.08, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{5}{2})}{0.08}$$

Solve for x (complex solution)

$$x=\frac{25\ln(5)}{2}-\frac{25\ln(2)}{2}+i\times 25\pi n_{1}$$

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