Solve 22=9^4x-3

$$22 = 9 ^ { 4 x - 3 }$$

Solve for x

$$x = \frac{\log_{3} {(22)} + 6}{8} \approx 1.101698512$$

Steps Using Definition of Logarithm

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

$$9^{4x-3}=22$$

Take the logarithm of both sides of the equation.

$$\log(9^{4x-3})=\log(22)$$

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

$$\left(4x-3\right)\log(9)=\log(22)$$

Divide both sides by \log(9).

$$4x-3=\frac{\log(22)}{\log(9)}$$

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

$$4x-3=\log_{9}\left(22\right)$$

Add 3 to both sides of the equation.

$$4x=\frac{\log_{3}\left(22\right)}{2}-\left(-3\right)$$

Divide both sides by 4.

$$x=\frac{\frac{\log_{3}\left(22\right)}{2}+3}{4}$$

Solve for x (complex solution)

$$x=\frac{\pi n_{1}i}{4\ln(3)}+\frac{\log_{3}\left(22\right)}{8}+\frac{3}{4}$$

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