# Solving a complex logarithmic equation

I am trying to solve the following equation for xx, with integer nn.

(1 – ((-1 + x)^E x^(1 – x) Log[x]^-E)^(1/(-1 + E)))/(-1 + x) == n

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Is xx constrained to be real? positive? non-negative? Can nn be zero or negative? You have not specified such constraints and I wish to verify that this is intentional. For instance, non-positive xx requires some complexity for evaluating Log[x].
– Eric Towers
May 12 ’15 at 23:46

Also, are you sure E belongs in the exponent? It’s not wrong persay, just a little bit unusual.
– 2012rcampion
May 13 ’15 at 2:59

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1

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Such an equation cannot be solved symbolically by Solve, but a solution might be approximately determined with FindRoot.

Looking at the left-hand side:

Plot[(1 – ((-1 + x)^E x^(1 – x) Log[x]^-E)^(1/(-1 + E)))/(-1 + x), {x, 0, 10}]

which has a (real) domain of x > 0, we see that the only integers for which the equation is likely to have a solution are n == -1, 0.

Table[
FindRoot[(1 – ((-1 + x)^E x^(1 – x) Log[x]^-E)^(1/(-1 + E)))/(-1 + x) == n,
{x, 2}],
{n, -1, 0}]
(* {{x -> 0.579149 + 7.4303*10^-20 I}, {x -> 2.46043}} *)

The imaginary part is negligible.