Solve 5000=2500e^.035x

$$5000=2500 { e }^{ .035x }$$

Solve for x

$$x=\frac{200\ln(2)}{7}\approx 19.804205159$$

Steps Using Definition of Logarithm

Divide both sides by 2500.

$$\frac{5000}{2500}=e^{0.035x}$$

Divide 5000 by 2500 to get 2.

$$2=e^{0.035x}$$

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

$$e^{0.035x}=2$$

Take the logarithm of both sides of the equation.

$$\log(e^{0.035x})=\log(2)$$

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

$$0.035x\log(e)=\log(2)$$

Divide both sides by \log(e).

$$0.035x=\frac{\log(2)}{\log(e)}$$

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

$$0.035x=\log_{e}\left(2\right)$$

Divide both sides of the equation by 0.035, which is the same as multiplying both sides by the reciprocal of the fraction.

$$x=\frac{\ln(2)}{0.035}$$

Solve for x (complex solution)

$$x=\frac{i\times 400\pi n_{1}}{7}+\frac{200\ln(2)}{7}$$

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