# Cannot solve for an unknown in an equation with exponents

I have an equation I need to solve for t. I have tried the code below:

Solve[
51684.2/342077 * 0.048 / 0.148 * 26.09/22.19 ==
Exp[t/26.09] - Exp[(22.19 t - 26.09 t) / 22.19 * 26.09],
t]


Wolfram Mathematica keeps calculating forever and never finishes. I have asked for help on the general stack exchange before (not the Mathematica section). They recommended two lines, both of which give a result, but neither is correct. When I manually substitute the value produced by the coded below, they do not satisfy the equation:

a = 51684.2/342077 * 0.048/0.148 * 26.09/22.19;
b = Exp[(22.19 t - 26.09 t)/22.19 * 26.09];
FindRoot[a == Exp[t/26.09] - b, {t, 0}]

{t -> 0.0128273}


OR

NSolve[a == Exp[t/26.09] - b, t, Reals]


I believe the answer should be in the the region of 10 – 100, absolutely not below 3.35.

• The value you found with FindRoot does satisfy the equation. eq = 51684.2/342077*0.048/0.148*26.09/22.19 == Exp[t/26.09] - Exp[(22.19 t - 26.09 t)/22.19*26.09]; sol = FindRoot[eq, {t, 0}]; eq /. sol yields True – MelaGo May 17 '19 at 4:10
• Plot the equation Plot[Exp[t/26.09] - Exp[(22.19 t - 26.09 t)/22.19*26.09] - 51684.2/342077*0.048/0.148*26.09/22.19, {t, 0, 100}] to check your belief. – JimB May 17 '19 at 4:13

NMinimize[( (51684.2/342077*0.048/0.148*26.09/22.19) -

If you are looking for a real solution you have to tell it NSolve:
NSolve[51684.2/342077*0.048/0.148*26.09/22.19 ==(Exp[t/26.09] -Exp[(22.19 t - 26.09 t)/22.19*26.09]), t, Reals]