Solve 25^2-x=125

$$25 ^ { 2 - x } = 125$$

Solve for x

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

Steps Using Definition of Logarithm

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

$$25^{-x+2}=125$$

Take the logarithm of both sides of the equation.

$$\log(25^{-x+2})=\log(125)$$

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

$$\left(-x+2\right)\log(25)=\log(125)$$

Divide both sides by \log(25).

$$-x+2=\frac{\log(125)}{\log(25)}$$

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

$$-x+2=\log_{25}\left(125\right)$$

Subtract 2 from both sides of the equation.

$$-x=\frac{3}{2}-2$$

Divide both sides by -1.

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

Solve for x (complex solution)

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

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