If I enter the equation in the following form, it does not solve it:
solve(exp(a * x) - C * exp(b * x)=0,x);
[%e^(b*x)=%e^(a*x)/C]
But if I rearrange the equation like this
solve(exp(a * x) /exp(b * x) = C,x);
[x=-log(C)/(b-a)]
then it gives the correct answer.