What is the natural log of -0.2877?

1 Answer
George C.
Jun 26, 2016

If you are talking about #ln# as a Real valued function of Real numbers then #ln(-0.2877)# is undefined.

The principal Complex natural logarithm is:

#ln(-0.2877) = ln(0.2877)+pii ~~ -1.2458+pii#

Explanation:

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Real natural logarithm

The function #e^x# with domain #(-oo, oo)# and range #(0, oo)# is one to one. So it has a well defined inverse function #ln(x)# with domain #(0, oo)# and range #(-oo, oo)#

Since negative numbers (and zero) are not in the range of #e^x#, they are not in the domain of the inverse Real logarithm function #ln(x)#. So #ln(-0.2877)# is undefined.

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Complex natural logarithm

The function #e^z# has domain #CC# and range #CC "\" { 0 }#

It is many to one, e.g. #e^0 = 1 = e^(2pii)#, so it does not have a well defined inverse function. However, if we restrict the domain to #{ z in CC : -pi < Im(z) <= pi }# then it has a well defined inverse:

#ln(z) = ln abs(z) + Arg(z)i#

In particular, for negative Real numbers, we have:

#ln(x) = ln(-x) + pii#

Hence:

#ln(-0.2877) = ln(0.2877)+pii ~~ -1.2458+pii#

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