Solve 5^x-8-4=52

$$5 ^ { x - 8 } - 4 = 52$$

Solve for x

$$x=\log_{5}\left(56\right)+8\approx 10.501091629$$

Steps Using Definition of Logarithm

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

$$5^{x-8}-4=52$$

Add 4 to both sides of the equation.

$$5^{x-8}=56$$

Take the logarithm of both sides of the equation.

$$\log(5^{x-8})=\log(56)$$

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

$$\left(x-8\right)\log(5)=\log(56)$$

Divide both sides by \log(5).

$$x-8=\frac{\log(56)}{\log(5)}$$

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

$$x-8=\log_{5}\left(56\right)$$

Add 8 to both sides of the equation.

$$x=\log_{5}\left(56\right)-\left(-8\right)$$

Solve for x (complex solution)

$$x=\frac{i\times 2\pi n_{1}}{\ln(5)}+\log_{5}\left(56\right)+8$$

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