How do you solve e^x=0?

1 Answer
Jun 30, 2016

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