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.