Solve 81^a=27

$$81 ^ { a } = 27$$

Solve for a

$$a=\frac{3}{4}=0.75$$

Steps Using Definition of Logarithm

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

$$81^{a}=27$$

Take the logarithm of both sides of the equation.

$$\log(81^{a})=\log(27)$$

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

$$a\log(81)=\log(27)$$

Divide both sides by \log(81).

$$a=\frac{\log(27)}{\log(81)}$$

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

$$a=\log_{81}\left(27\right)$$

Solve for a (complex solution)

$$a=\frac{\pi n_{1}i}{2\ln(3)}+\frac{3}{4}$$

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