Solve 9^c=243

$$9 ^ { c } = 243$$

Solve for c

$$c = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using Definition of Logarithm

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

$$9^{c}=243$$

Take the logarithm of both sides of the equation.

$$\log(9^{c})=\log(243)$$

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

$$c\log(9)=\log(243)$$

Divide both sides by \log(9).

$$c=\frac{\log(243)}{\log(9)}$$

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

$$c=\log_{9}\left(243\right)$$

Solve for c (complex solution)

$$c=\frac{\pi n_{1}i}{\ln(3)}+\frac{5}{2}$$

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