Solve 9^n*3^5*27^3/2*21^4=27

$$\frac { 9 ^ { n } \times 3 ^ { 5 } \times 27 ^ { 3 } } { 2 \times 21 ^ { 4 } } = 27$$

Solve for n

$$n=\frac{\log_{3}\left(4802\right)-7}{2}\approx 0.357952375$$

Steps Using Definition of Logarithm

Calculate 3 to the power of 5 and get 243.

$$\frac{9^{n}\times 243\times 27^{3}}{2\times 21^{4}}=27$$

Calculate 27 to the power of 3 and get 19683.

$$\frac{9^{n}\times 243\times 19683}{2\times 21^{4}}=27$$

Multiply 243 and 19683 to get 4782969.

$$\frac{9^{n}\times 4782969}{2\times 21^{4}}=27$$

Calculate 21 to the power of 4 and get 194481.

$$\frac{9^{n}\times 4782969}{2\times 194481}=27$$

Multiply 2 and 194481 to get 388962.

$$\frac{9^{n}\times 4782969}{388962}=27$$

Divide 9^{n}\times 4782969 by 388962 to get 9^{n}\times \frac{59049}{4802}.

$$9^{n}\times \frac{59049}{4802}=27$$

Multiply both sides by \frac{4802}{59049}, the reciprocal of \frac{59049}{4802}.

$$9^{n}=27\times \frac{4802}{59049}$$

Multiply 27 and \frac{4802}{59049} to get \frac{4802}{2187}.

$$9^{n}=\frac{4802}{2187}$$

Take the logarithm of both sides of the equation.

$$\log(9^{n})=\log(\frac{4802}{2187})$$

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

$$n\log(9)=\log(\frac{4802}{2187})$$

Divide both sides by \log(9).

$$n=\frac{\log(\frac{4802}{2187})}{\log(9)}$$

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

$$n=\log_{9}\left(\frac{4802}{2187}\right)$$