Solve 3^2-4=9^x-2

$$3 ^ { 2 - 4 } = 9 ^ { x - 2 }$$

Solve for x

$$x=1$$

Steps Using Definition of Logarithm

Subtract 4 from 2 to get -2.

$$3^{-2}=9^{x-2}$$

Calculate 3 to the power of -2 and get \frac{1}{9}.

$$\frac{1}{9}=9^{x-2}$$

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

$$9^{x-2}=\frac{1}{9}$$

Take the logarithm of both sides of the equation.

$$\log(9^{x-2})=\log(\frac{1}{9})$$

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

$$\left(x-2\right)\log(9)=\log(\frac{1}{9})$$

Divide both sides by \log(9).

$$x-2=\frac{\log(\frac{1}{9})}{\log(9)}$$

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

$$x-2=\log_{9}\left(\frac{1}{9}\right)$$

Add 2 to both sides of the equation.

$$x=-1-\left(-2\right)$$

Solve for x (complex solution)

$$x=\frac{i\pi n_{1}}{\ln(3)}+1$$

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