Solve .5^x=15,625

$$.5 ^ { x } = 15,625$$

Solve for x

$$x=-6\log_{2}\left(5\right)\approx -13.931568569$$

Steps Using Definition of Logarithm

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

$$0.5^{x}=15625$$

Take the logarithm of both sides of the equation.

$$\log(0.5^{x})=\log(15625)$$

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

$$x\log(0.5)=\log(15625)$$

Divide both sides by \log(0.5).

$$x=\frac{\log(15625)}{\log(0.5)}$$

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

$$x=\log_{0.5}\left(15625\right)$$

Solve for x (complex solution)

$$x=-\frac{i\times 2\pi n_{1}}{\ln(2)}-6\log_{2}\left(5\right)$$

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