Is it true that #ln x# is the inverse of the function #e^x# ?

1 Answer
Oct 19, 2017

True

Explanation:

As a real valued function of real numbers, #e^x# is a one to one, continuous, strictly monotonically increasing function from #(-oo, oo)# onto #(0, oo)#.

Its inverse is the real natural logarithm #ln x#, which is a one to one, continuous, strictly monotonically increasing function from #(0, oo)# onto #(-oo, oo)#.

So for any #x in (0, oo)# we have:

#e^(ln(x)) = x#

and for any #x in (-oo, oo)# we have:

#ln e^x = x#