# Solving $$b^n +nc+d\leq 0$$

Can someone give me a hint how to solve the inequality $$b^n +nc+d\leq 0$$

for $n\in \mathbb{N}$, where $b,c,d\in \mathbb{R}$ and $-1\leq d\leq 0,\ c\leq -1$ and $c\geq 2$?

I think I need some sort of "inverse function" that enables me to remove $n$ from the exponent, but using logarithms didn't help.

I am more interested in a general way/strategy of solving this type of inequalities that doesn't tie me these special values of $b,c,d$.

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 do you have information about b, n and d? – Emmad Kareem Mar 31 '12 at 9:26 You can solve analytically $b^n=-nc-d$ for $n$ only in terms of Lambert W function... – pedja Mar 31 '12 at 9:35 @EmmadKareem Yes, I updated the question – user10324 Mar 31 '12 at 9:35 @pedja But isn't there a way to solve it without resorting to complex analysis ? (Because I'm not very good at that) Or isn't there a real analogue of the Lamberts W function ? – user10324 Mar 31 '12 at 9:44 – draks ... Mar 31 '12 at 10:42