Solve 3^8=9^n*6^4

$$3 ^ { 8 } = 9 ^ { n } \cdot 6 ^ { 4 }$$

Solve for n

$$n=-2\log_{3}\left(2\right)+2\approx 0.738140493$$

Steps Using Definition of Logarithm

Calculate 3 to the power of 8 and get 6561.

$$6561=9^{n}\times 6^{4}$$

Calculate 6 to the power of 4 and get 1296.

$$6561=9^{n}\times 1296$$

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

$$9^{n}\times 1296=6561$$

Divide both sides by 1296.

$$9^{n}=\frac{6561}{1296}$$

Reduce the fraction \frac{6561}{1296} to lowest terms by extracting and canceling out 81.

$$9^{n}=\frac{81}{16}$$

Take the logarithm of both sides of the equation.

$$\log(9^{n})=\log(\frac{81}{16})$$

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

$$n\log(9)=\log(\frac{81}{16})$$

Divide both sides by \log(9).

$$n=\frac{\log(\frac{81}{16})}{\log(9)}$$

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

$$n=\log_{9}\left(\frac{81}{16}\right)$$