why does e^(-x) = 1 have at least one solution?

Question

(i know this is so annoying)

thnx for your help!

Answer 1

How about why does it have only one solution?

Suppose (a,b) were both solutions with a<>b
Then
e^(-a)=e^(-b)
and
ln(e^(-a))=ln(e^(-b))
so
-a = -b
and
a = b contradiction!

Since x=0 is a solution, it is therefore the only solution.

Answer 2

Because of that other annoying question, why is x^0=1 for all x?

Answer 3

Because...

when x = 0: e^(-x) = e^0 = 1

If you change your equation around: e^(-x)-1=0 and plot it you will see that it crosses the X-Axis in at least one place.