Solve 25=625^x

$$25 = 625 ^ { x }$$

Solve for x

$$x=\frac{1}{2}=0.5$$

Steps Using Definition of Logarithm

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

$$625^{x}=25$$

Take the logarithm of both sides of the equation.

$$\log(625^{x})=\log(25)$$

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

$$x\log(625)=\log(25)$$

Divide both sides by \log(625).

$$x=\frac{\log(25)}{\log(625)}$$

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

$$x=\log_{625}\left(25\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{2\ln(5)}+\frac{1}{2}$$

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