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\frac{3^{4}\times 27^{3}\times 9^{n}}{81^{4}}=27
Cancel out 3 in both numerator and denominator.
\frac{81\times 27^{3}\times 9^{n}}{81^{4}}=27
Calculate 3 to the power of 4 and get 81.
\frac{81\times 19683\times 9^{n}}{81^{4}}=27
Calculate 27 to the power of 3 and get 19683.
\frac{1594323\times 9^{n}}{81^{4}}=27
Multiply 81 and 19683 to get 1594323.
\frac{1594323\times 9^{n}}{43046721}=27
Calculate 81 to the power of 4 and get 43046721.
\frac{1}{27}\times 9^{n}=27
Divide 1594323\times 9^{n} by 43046721 to get \frac{1}{27}\times 9^{n}.
9^{n}=27\times 27
Multiply both sides by 27, the reciprocal of \frac{1}{27}.
9^{n}=729
Multiply 27 and 27 to get 729.
\log(9^{n})=\log(729)
Take the logarithm of both sides of the equation.
n\log(9)=\log(729)
The logarithm of a number raised to a power is the power times the logarithm of the number.
n=\frac{\log(729)}{\log(9)}
Divide both sides by \log(9).
n=\log_{9}\left(729\right)
By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).