What is the natural log of -0.2877?

Question

Madelyn Anderson · Accepted Answer

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#

#color(white)()# 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. #color(white)()# 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#