Solve 125=25^x+3

$$125 = 25 ^ { x + 3 }$$

Solve for x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using Definition of Logarithm

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

$$25^{x+3}=125$$

Take the logarithm of both sides of the equation.

$$\log(25^{x+3})=\log(125)$$

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

$$\left(x+3\right)\log(25)=\log(125)$$

Divide both sides by \log(25).

$$x+3=\frac{\log(125)}{\log(25)}$$

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

$$x+3=\log_{25}\left(125\right)$$

Subtract 3 from both sides of the equation.

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

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(5)}-\frac{3}{2}$$

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