I have two exponential functions, how can I generate any exponential function within the range of the two functions?

I have two exponential function like: A1(1/λ)^k1, and A2(1/λ)^k2 how can I find any exponential function within the range of the two functions? The function we need also follow this form. For example: Ax(1/λ)^kx, which Ax and kx are the value I want to know.

Here is my pseudo code:

def fun_random_gen(fun_upper_bound, fun_lower_bound)
    generate Ax(1/λ)^kx within fun_upper_bound, fun_lower_bound
    return Ax, kx

Appreciate !