Question

Are exponential functions necessarily one to one?

Answer 1

If `a, b in RR` and `a, b > 0`, then `f(x) = ab^x` is one-one from `RR` onto `(0, oo)`

Apart from such normal cases, exponential functions are often not one-one.

Explanation:

As a Complex valued function from `CC->CC` \ `{ 0 }`, the exponential function `exp(z) = e^z` is many to one.

For example, `e^(2k pi i) = 1` for all `k in ZZ`