Question

How do you solve `e^x=0`?

Answer 1

There is no `x` such that `e^x = 0`

Explanation:

The function `e^x` considered as a function of Real numbers has domain `(-oo, oo)` and range `(0, oo)`.

So it can only take strictly positive values.

When we consider `e^x` as a function of Complex numbers, then we find it has domain `CC` and range `CC "\" { 0 }`.

That is `0` is the only value that `e^x` cannot take.

Note that `e^(x+yi) = e^x e^(yi) = e^x(cos y+i sin y)`

We have already noted that iof `x in RR` then `e^x > 0`.

For pure imaginary exponents the result is on the unit circle, specifically:

`e^(yi) = cos y + i sin y != 0`

So `e^(x+yi) != 0` for all `x, y in RR`