# Solving two equations with logarithmic terms with NSolve

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.

• What a priori knowledge to have that makes you believe there is a real solution to this system? – m_goldberg Jan 9 at 5:34

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 $$6.6612$$.