Solve sqrt{81}=3^x

$$\sqrt{ 81 } = { 3 }^{ x }$$

Solve for x

$$x=2$$

Steps Using Definition of Logarithm

Calculate the square root of 81 and get 9.

$$9=3^{x}$$

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

$$3^{x}=9$$

Take the logarithm of both sides of the equation.

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

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

$$x\log(3)=\log(9)$$

Divide both sides by \log(3).

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

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

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

Solve for x (complex solution)

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

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