Solve 0.125=2^x

$$0.125= { 2 }^{ x }$$

Solve for x

$$x=-3$$

Steps Using Definition of Logarithm

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

$$2^{x}=0.125$$

Take the logarithm of both sides of the equation.

$$\log(2^{x})=\log(0.125)$$

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

$$x\log(2)=\log(0.125)$$

Divide both sides by \log(2).

$$x=\frac{\log(0.125)}{\log(2)}$$

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

$$x=\log_{2}\left(0.125\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(2)}-3$$

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