Can the logarithm of a number be negative? Can it be imaginary?

1 Answer
Oct 31, 2015

Answer:

Yes and yes, but it gets complicated...

Explanation:

#e^x:RR->(0,oo)# is a one to one Real-valued function with inverse #ln:(0,oo)->RR#

If #0 < x < 1# then #ln x < 0#

#e^z:CC->CC \\ {0}# is a many to one Complex-valued function. As a result, it has no inverse function. However, it is possible to extend the definition of logs to Complex numbers. We can limit the domain of #e^z# to make it a one to one function, allowing the definition of an inverse "#ln z#".

For example,

#e^z:{a+ib in CC : -pi < b <= pi} -> CC \\ {0}# is a one one function.

If we use this definition then

#ln z:CC \\ {0} -> {a+ib in CC : -pi < b <= pi}#

is well defined and we find values like #ln(-1) = i pi#