Solve 250=100e^0.09*x

$$250 = 100 e ^ { 0.09 \cdot x }$$

Solve for x

$$x=\frac{100\ln(5)-100\ln(2)}{9}\approx 10.181008132$$

Steps Using Definition of Logarithm

Divide both sides by 100.

$$\frac{250}{100}=e^{0.09x}$$

Reduce the fraction \frac{250}{100} to lowest terms by extracting and canceling out 50.

$$\frac{5}{2}=e^{0.09x}$$

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

$$e^{0.09x}=\frac{5}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.09x})=\log(\frac{5}{2})$$

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

$$0.09x\log(e)=\log(\frac{5}{2})$$

Divide both sides by \log(e).

$$0.09x=\frac{\log(\frac{5}{2})}{\log(e)}$$

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

$$0.09x=\log_{e}\left(\frac{5}{2}\right)$$

Divide both sides of the equation by 0.09, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{5}{2})}{0.09}$$

Solve for x (complex solution)

$$x=\frac{i\times 200\pi n_{1}}{9}+\frac{100\ln(5)}{9}-\frac{100\ln(2)}{9}$$

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