Solve (0.05)^x=0.125

$$( 0.05 ) ^ { x } = 0.125$$

Solve for x

$$x=3\log_{20}\left(2\right)\approx 0.694134639$$

Steps Using Definition of Logarithm

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

$$0.05^{x}=0.125$$

Take the logarithm of both sides of the equation.

$$\log(0.05^{x})=\log(0.125)$$

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

$$x\log(0.05)=\log(0.125)$$

Divide both sides by \log(0.05).

$$x=\frac{\log(0.125)}{\log(0.05)}$$

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

$$x=\log_{0.05}\left(0.125\right)$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{\ln(20)}+3\log_{20}\left(2\right)$$

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