Solve 500=200*1.03^2x

$$500 = 200 \times 1.03 ^ { 2 x }$$

Solve for x

$$x = \frac{\log_{1.03} {(2.5)}}{2} \approx 15.499456378$$

Steps Using Definition of Logarithm

Divide both sides by 200.

$$\frac{500}{200}=1.03^{2x}$$

Reduce the fraction \frac{500}{200} to lowest terms by extracting and canceling out 100.

$$\frac{5}{2}=1.03^{2x}$$

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

$$1.03^{2x}=\frac{5}{2}$$

Take the logarithm of both sides of the equation.

$$\log(1.03^{2x})=\log(\frac{5}{2})$$

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

$$2x\log(1.03)=\log(\frac{5}{2})$$

Divide both sides by \log(1.03).

$$2x=\frac{\log(\frac{5}{2})}{\log(1.03)}$$

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

$$2x=\log_{1.03}\left(\frac{5}{2}\right)$$

Divide both sides by 2.

$$x=\frac{\ln(\frac{5}{2})}{2\ln(\frac{103}{100})}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(1.03)}+\frac{\log_{1.03}\left(2.5\right)}{2}$$

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