$$\frac { 9 ^ { 7 } \times 3 ^ { 2 } \times 3 ^ { n } - ( 27 ) ^ { n } } { ( 3 ^ { 3 } ) ^ { 5 } \times 2 ^ { 3 } } = \frac { 1 } { 27 }$$
$n=\log_{3}\left(2187\sqrt{3}\left(\sqrt{3}\sin(\frac{\arccos(\frac{4\sqrt{3}}{177147})}{3})+\cos(\frac{\arccos(\frac{4\sqrt{3}}{177147})}{3})\right)\right)\approx 7.999993149$
$n=\log_{3}\left(2187\left(\sqrt{3}\cos(\frac{\arccos(\frac{4\sqrt{3}}{177147})}{3})-3\sin(\frac{\arccos(\frac{4\sqrt{3}}{177147})}{3})\right)\right)\approx -2.107210739$
