Hello Turda,

I believe that a numerical solution is the only way to go with this one.

Here is a method with the correct syntax (it should be a good excercise for you to figure out the what the syntax is doing from the results). I've put in several steps here so you can see what is going on. It will more instructive if you evaluate this one statement at a time.

gc = a Exp[b c] - a

eqs = {1 == gc /. c -> 1/10, 100 == gc /. c -> 100 }

Simplify[eqs[[2]] /. Solve[eqs[[1]], a]]

Plot[100 - (-1 + Exp[100 b])/(-1 + Exp[b/10]), {b, -0.2, -0.05}, PlotRange -> {-1, 1}]

FindRoot[100 - (-1 + E^(100 b))/(-1 + E^(b/10)), {b, -0.1}]

(*{b -> -0.100499}*)

You can find the value of a by using the rule: b -> -0.100499

Because the function changes so rapidly around values of b=0, it is helpful to plot the function so that you can give the numerical root finding technique a bit of help.