To Solve Questions of this Type
There is no algebraic method to solve equations like this. If the number on the right hand side had been 91853 instead then the answer can not be written in terms of common expressions (trigonometry, logs/exponentials, arithmetic, etc). Generally questions of this type have to be answered numerically - which could include a number of ways: trial and error, computer software, graphing, etc.
To Solve this Specific Question (or ones where the answer is an integer)
Try to factorize 91854. You'll find that:
The $7$ stands out as compared to the other numbers. As we are assuming integer solutions it must be used one of the parts on the left or a combination of them.
Some are easily excluded as they clearly do not have integer solutions. E.g.: $3^n=7\times something$, $n\times3^n=7\times something$, $(n-1)3^n=7\times something$, etc.
The one which looks potentially like a solution is: $n(n-1)=7\times something$
So you can conclude that either $n=7$ or $n-1=7$. Checking easily reveals that the answer is $n=7$.