Solve 12^x-2=20

$$12 ^ { x - 2 } = 20$$

Solve for x

$$x=\log_{12}\left(2880\right)\approx 3.205571354$$

Steps Using Definition of Logarithm

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

$$12^{x-2}=20$$

Take the logarithm of both sides of the equation.

$$\log(12^{x-2})=\log(20)$$

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

$$\left(x-2\right)\log(12)=\log(20)$$

Divide both sides by \log(12).

$$x-2=\frac{\log(20)}{\log(12)}$$

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

$$x-2=\log_{12}\left(20\right)$$

Add 2 to both sides of the equation.

$$x=\log_{12}\left(20\right)-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{2\pi n_{1}i}{\ln(12)}+\log_{12}\left(2880\right)$$

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