Solve frac{9^{n}*3^{2}*3^{n}-27^n}{(3^3)^5*2^3}=1/27

$$\frac { 9 ^ { n } \times 3 ^ { 2 } \times 3 ^ { n } - 27 ^ { n } } { ( 3 ^ { 3 } ) ^ { 5 } \times 2 ^ { 3 } } = \frac { 1 } { 27 }$$

Solve for n

$$n=4$$

Steps Using Definition of Logarithm

To raise a power to another power, multiply the exponents. Multiply 3 and 5 to get 15.

$$\frac{9^{n}\times 3^{2}\times 3^{n}-27^{n}}{3^{15}\times 2^{3}}=\frac{1}{27}$$

Calculate 3 to the power of 2 and get 9.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{3^{15}\times 2^{3}}=\frac{1}{27}$$

Calculate 3 to the power of 15 and get 14348907.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{14348907\times 2^{3}}=\frac{1}{27}$$

Calculate 2 to the power of 3 and get 8.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{14348907\times 8}=\frac{1}{27}$$

Multiply 14348907 and 8 to get 114791256.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256}=\frac{1}{27}$$

Subtract \frac{1}{27} from both sides.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256}-\frac{1}{27}=0$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 114791256 and 27 is 114791256. Multiply \frac{1}{27} times \frac{4251528}{4251528}.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256}-\frac{4251528}{114791256}=0$$

Since \frac{9^{n}\times 9\times 3^{n}-27^{n}}{114791256} and \frac{4251528}{114791256} have the same denominator, subtract them by subtracting their numerators.

$$\frac{9^{n}\times 9\times 3^{n}-27^{n}-4251528}{114791256}=0$$

Do the multiplications in 9^{n}\times 9\times 3^{n}-27^{n}-4251528.

$$\frac{9\times 27^{n}-27^{n}-4251528}{114791256}=0$$

Combine like terms in 9\times 27^{n}-27^{n}-4251528.

$$\frac{8\times 27^{n}-4251528}{114791256}=0$$

Divide each term of 8\times 27^{n}-4251528 by 114791256 to get \frac{1}{14348907}\times 27^{n}-\frac{1}{27}.

$$\frac{1}{14348907}\times 27^{n}-\frac{1}{27}=0$$

Add \frac{1}{27} to both sides of the equation.

$$\frac{1}{14348907}\times 27^{n}=\frac{1}{27}$$

Multiply both sides by 14348907.

$$27^{n}=531441$$

Take the logarithm of both sides of the equation.

$$\log(27^{n})=\log(531441)$$

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

$$n\log(27)=\log(531441)$$

Divide both sides by \log(27).

$$n=\frac{\log(531441)}{\log(27)}$$

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

$$n=\log_{27}\left(531441\right)$$