Solve 8=5^x-2

$$8 = 5 ^ { x - 2 }$$

Solve for x

$$x=3\log_{5}\left(2\right)+2\approx 3.292029674$$

Steps Using Definition of Logarithm

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

$$5^{x-2}=8$$

Take the logarithm of both sides of the equation.

$$\log(5^{x-2})=\log(8)$$

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

$$\left(x-2\right)\log(5)=\log(8)$$

Divide both sides by \log(5).

$$x-2=\frac{\log(8)}{\log(5)}$$

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

$$x-2=\log_{5}\left(8\right)$$

Add 2 to both sides of the equation.

$$x=3\log_{5}\left(2\right)-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(5)}+3\log_{5}\left(2\right)+2$$

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