Solve 22(1/3)^x-7=22

$$22 ( 1 / 3 ) ^ { x - 7 } = 22$$

Solve for x

$$x=7$$

Steps Using Definition of Logarithm

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

$$22\times \left(\frac{1}{3}\right)^{x-7}=22$$

Divide both sides by 22.

$$\left(\frac{1}{3}\right)^{x-7}=1$$

Take the logarithm of both sides of the equation.

$$\log(\left(\frac{1}{3}\right)^{x-7})=\log(1)$$

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

$$\left(x-7\right)\log(\frac{1}{3})=\log(1)$$

Divide both sides by \log(\frac{1}{3}).

$$x-7=\frac{\log(1)}{\log(\frac{1}{3})}$$

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

$$x-7=\log_{\frac{1}{3}}\left(1\right)$$

Add 7 to both sides of the equation.

$$x=-\left(-7\right)$$

Solve for x (complex solution)

$$x=-\frac{2\pi n_{1}i}{\ln(3)}+7$$

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