Eqn = exp(-0.04*t)+exp(-0.12*t) - 1;
Write it like that. I did so because now we can plot it. See that I specified t to be a real variable, since you are not interested in complex valued solutions.
Now, we can plot it. Does a solution exist? Are there multiple solutions? ALWAYS PLOT EVERYTHING. Look at what you see. Think about what you see. Then plot it in a different way if necessary. Where this relationship crosses zero, this is your solution.
So a solution does exist, and I would bet there is only one real solution. The one you care about is a little less than 10.
tsol = solve(Eqn,t)
tsol = 
And that may not seem terribly useful, but it is. The result is the third root of a cubic polynomial, You need to push MATLAB to resolve the solution. VPA will do that here.
vpa(tsol)
ans = 9.5561271460008910332339624796214
You could also have gone directly to a numerical solution using vpasolve.
vpasolve(Eqn)
ans = 9.5561271460008910332339624796214
In fact, that polynomial had three roots, but two will yield complex values. If you wish an algebraic solution for the problem, you could do this:
solve(Eqn,t,'maxdegree',3)
ans =
