Solve (0.1)^2x=100

$$( 0.1 ) ^ { 2 x } = 100$$

Solve for x

$$x=-1$$

Steps Using Definition of Logarithm

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

$$0.1^{2x}=100$$

Take the logarithm of both sides of the equation.

$$\log(0.1^{2x})=\log(100)$$

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

$$2x\log(0.1)=\log(100)$$

Divide both sides by \log(0.1).

$$2x=\frac{\log(100)}{\log(0.1)}$$

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

$$2x=\log_{0.1}\left(100\right)$$

Divide both sides by 2.

$$x=-\frac{2}{2}$$

Solve for x (complex solution)

$$x=-i\pi n_{1}\log(e)-1$$

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