Solve 10000=1500e^0.15x

$$10000 = 1500 e ^ { 0.15 x }$$

Solve for x

$$x=\frac{20\ln(20)-20\ln(3)}{3}\approx 12.647466566$$

Steps Using Definition of Logarithm

Divide both sides by 1500.

$$\frac{10000}{1500}=e^{0.15x}$$

Reduce the fraction \frac{10000}{1500} to lowest terms by extracting and canceling out 500.

$$\frac{20}{3}=e^{0.15x}$$

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

$$e^{0.15x}=\frac{20}{3}$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.15x})=\log(\frac{20}{3})$$

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

$$0.15x\log(e)=\log(\frac{20}{3})$$

Divide both sides by \log(e).

$$0.15x=\frac{\log(\frac{20}{3})}{\log(e)}$$

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

$$0.15x=\log_{e}\left(\frac{20}{3}\right)$$

Divide both sides of the equation by 0.15, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{20}{3})}{0.15}$$

Solve for x (complex solution)

$$x=\frac{i\times 40\pi n_{1}}{3}+\frac{20\ln(20)}{3}-\frac{20\ln(3)}{3}$$

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