Solve 3^4(81^2)=3^n

$$3 ^ { 4 } ( 81 ^ { 2 } ) = 3 ^ { n }$$

Solve for n

$$n=12$$

Steps Using Definition of Logarithm

Calculate 3 to the power of 4 and get 81.

$$81\times 81^{2}=3^{n}$$

Calculate 81 to the power of 2 and get 6561.

$$81\times 6561=3^{n}$$

Multiply 81 and 6561 to get 531441.

$$531441=3^{n}$$

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

$$3^{n}=531441$$

Take the logarithm of both sides of the equation.

$$\log(3^{n})=\log(531441)$$

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

$$n\log(3)=\log(531441)$$

Divide both sides by \log(3).

$$n=\frac{\log(531441)}{\log(3)}$$

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

$$n=\log_{3}\left(531441\right)$$

Solve for n (complex solution)

$$n=\frac{2\pi n_{1}i}{\ln(3)}+12$$

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