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I want to solve the following set of equations for $M$ and $\epsilon$ :

16.93963 == Log[1 + 1.15022 M ϵ]/ϵ
6.6612 == (0.8694/ϵ + M) Log[0.8694/ϵ + M] - (0.8694/ϵ) Log[0.8694/ϵ]

I used NSolve:

 NSolve[
   {6.6612 == (0.8694/ϵ + M) Log[0.8694/ϵ + M] - (0.8694/ϵ) Log[0.8694/ϵ], 
    1/0.1441 == 1/ϵ Log[1 + M ϵ/0.8694 ]}, {ϵ, M}, Reals]

It has been running for quite some time now and not yet returning a solution. Is there some clever trick to get the solution for this system of equations?

Update

The output came and it was same as the input.

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    $\begingroup$ What a priori knowledge to have that makes you believe there is a real solution to this system? $\endgroup$ – m_goldberg Jan 9 at 5:34
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Solve the first equation for $M$, and plug that in to the RHS of the second equation. You get an expression that, for positive $ϵ$, bottoms out around 85 (with $ϵ$ near $0.02$. In particular, it never gets anywhere near $6.6612$.

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