# Math Help - not-so-simple equation to solve

1. ## not-so-simple equation to solve

Dear all,
I have a tricky problem; I need to solve this equation for $N$

$\binom {L} {L-R}\sum_{r=1}^{R}\binom {R} {r} (-1)^{R-r}\left(\frac{r}{L}\right)^N\ln\frac{r}{L} =0
$

Is it possible in some way?
The only constraints are $R \leqslant L$, $R,L>0$

Thanks a lot
Simo

2. Going gets tough, tough resort to numerical methods. The following is Mathematica code to calculate the value of n for the supplied values of R and L. First out is the expression for the sum, second out is the numerically calculated value of n which makes the sum equal to zero, and the third out is a back-substitution of the the root for a check.

Code:
In[234]:=
R = 3;
L = 5;
formula1 = Sum[Binomial[R, r]*(-1)^(R - r)*
(r/L)^n*Log[r/L], {r, 1, R}]
root = n /. FindRoot[formula1 == 0,
{n, 1.2}]
formula1 /. n -> root

Out[236]= -(3/5)^n Log[5/3] + 3 (2/5)^n Log[5/2] - 3 5^-n Log[5]

Out[237]= 1.38015

Out[238]= -5.55112*10^-17

3. Ok,
I already have a numerical solution like that, but it is not interesting for me.
Is not possible to find an analytical solution, without resort to numerical methods?