Solve (100)(102^2x)=500000

$$(100)( { 102 }^{ 2x } )=500000$$

Solve for x

$$x=\frac{\log_{102}\left(5000\right)}{2}\approx 0.920783055$$

Steps Using Definition of Logarithm

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

$$100\times 102^{2x}=500000$$

Divide both sides by 100.

$$102^{2x}=5000$$

Take the logarithm of both sides of the equation.

$$\log(102^{2x})=\log(5000)$$

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

$$2x\log(102)=\log(5000)$$

Divide both sides by \log(102).

$$2x=\frac{\log(5000)}{\log(102)}$$

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

$$2x=\log_{102}\left(5000\right)$$

Divide both sides by 2.

$$x=\frac{\log_{102}\left(5000\right)}{2}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(102)}+\frac{\log_{102}\left(5000\right)}{2}$$

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