Solve 9^z=27

$$9 ^ { z } = 27$$

Solve for z

$$z = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

Steps Using Definition of Logarithm

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

$$9^{z}=27$$

Take the logarithm of both sides of the equation.

$$\log(9^{z})=\log(27)$$

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

$$z\log(9)=\log(27)$$

Divide both sides by \log(9).

$$z=\frac{\log(27)}{\log(9)}$$

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

$$z=\log_{9}\left(27\right)$$

Solve for z (complex solution)

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

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