What can be the value of $m$ in following equation

Question

During calculations I got this step $$(e^m/((m+1)^{m+1}) )^{3n/4} = 1/2^n$$

I want the value of m here??

Answer 1

Well, you can eliminate $n$ by raising both sides to $4/(3n)$:

$$\frac{e^m}{(m+1)^{m+1}} = \frac{1}{2^{4/3}}.$$

I don't think there's much hope of a closed-form solution, but Wolfram Alpha can easily find $m$ numerically.

Note that more, (likely complex, depending on $n$) solutions also exist -- multiply the right-hand side by $4/(3n)$th roots of unity.