Solve (9 ^ n * 3 ^ 2 * 3 ^ n - (27) ^ n)/((3 ^ 3) ^ 5 * 2 ^ 3) = 1/27 Find the value of n . | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

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

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 $\frac{\log(a)}{\log(b)}=\log_{b}\left(a\right)$.

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

Solve for n (complex solution)

$n=\frac{2\pi n_{1}i}{3\ln(3)}+4$
$n_{1}\in \mathrm{Z}$