Solve 1000e^0.0475x=1800

$$1000 { e }^{ 0.0475x } = 1800$$

Solve for x

$$x=\frac{800\ln(3)-400\ln(5)}{19}\approx 12.374456103$$

Steps Using Definition of Logarithm

Use the rules of exponents and logarithms to solve the equation.

$$1000e^{0.0475x}=1800$$

Divide both sides by 1000.

$$e^{0.0475x}=\frac{9}{5}$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.0475x})=\log(\frac{9}{5})$$

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

$$0.0475x\log(e)=\log(\frac{9}{5})$$

Divide both sides by \log(e).

$$0.0475x=\frac{\log(\frac{9}{5})}{\log(e)}$$

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

$$0.0475x=\log_{e}\left(\frac{9}{5}\right)$$

Divide both sides of the equation by 0.0475, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(\frac{9}{5})}{0.0475}$$

Solve for x (complex solution)

$$x=\frac{i\times 800\pi n_{1}}{19}+\frac{800\ln(3)}{19}-\frac{400\ln(5)}{19}$$

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