Solve 0.5^x=0.05

$$0.5 ^ { x } = 0.05$$

Solve for x

$$x=\log_{2}\left(20\right)\approx 4.321928095$$

Steps Using Definition of Logarithm

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

$$0.5^{x}=0.05$$

Take the logarithm of both sides of the equation.

$$\log(0.5^{x})=\log(0.05)$$

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

$$x\log(0.5)=\log(0.05)$$

Divide both sides by \log(0.5).

$$x=\frac{\log(0.05)}{\log(0.5)}$$

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

$$x=\log_{0.5}\left(0.05\right)$$

Solve for x (complex solution)

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

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