Solve (100)(1.02^2x)=500000

$$(100)( { 1.02 }^{ 2x } )=500000$$

Solve for x

$$x = \frac{\log_{1.02} {(5000)}}{2} \approx 215.052100512$$

Steps Using Definition of Logarithm

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

$$100\times 1.02^{2x}=500000$$

Divide both sides by 100.

$$1.02^{2x}=5000$$

Take the logarithm of both sides of the equation.

$$\log(1.02^{2x})=\log(5000)$$

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

$$2x\log(1.02)=\log(5000)$$

Divide both sides by \log(1.02).

$$2x=\frac{\log(5000)}{\log(1.02)}$$

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

$$2x=\log_{1.02}\left(5000\right)$$

Divide both sides by 2.

$$x=\frac{\ln(5000)}{2\ln(\frac{51}{50})}$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(1.02)}+\frac{\log_{1.02}\left(5000\right)}{2}$$

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