Solving an equation with a logarithm in the exponent

Hint: Use the rules of logarithm, especially the power rule and change of base rule.

Take $\log_6$ on both sides, and then simplify the equation to obtain $\log_6 5 = \log_{N+1} N$.

Observe that the graph of $\log_{N+1} N$ is monotonic (for example, by differentiating), hence the unique answer is $N=5$.

Note that $\;216 = 6^3,\;$ and $\;125 = 5^3,\;$ so use the logarithm power rule: $\;\log (a^b) = b \log a\;$ to write:

$$ (N+1)^{\large\log_N{ 5^3}} = (N+ 1)^{3\large\log_N5} = 6^3$$

That is $$\left [ (N+1)^3 \right ]^{\large\log_N{5}} = (5 + 1)^3 $$ and your solution is apparent.

You can rewrite the equation so the solution is evident, such as

$$\left [ (N+1)^3 \right ]^{\log_N{5}} = (5 + 1)^3 $$