Solve 9^23/9^11=9^x

$$\frac { 9 ^ { 23 } } { 9 ^ { 11 } } = 9 ^ { x }$$

Solve for x

$$x=12$$

Steps Using Definition of Logarithm

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent. Subtract 11 from 23 to get 12.

$$9^{12}=9^{x}$$

Calculate 9 to the power of 12 and get 282429536481.

$$282429536481=9^{x}$$

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

$$9^{x}=282429536481$$

Take the logarithm of both sides of the equation.

$$\log(9^{x})=\log(282429536481)$$

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

$$x\log(9)=\log(282429536481)$$

Divide both sides by \log(9).

$$x=\frac{\log(282429536481)}{\log(9)}$$

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

$$x=\log_{9}\left(282429536481\right)$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{\ln(3)}+12$$

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