What is the natural log of zero?

Question

Ethan Adkins · Accepted Answer

Difficult one!

This is a tricky question because you do not have a unique answer...I mean, you do not have an answer such as: "the result is 3". The problem here rests in the definition of log: #log_ax=b -> x=a^b# so basically with the log you are looking for a certain exponent that when you rise the base to it gives you the integrand. Now, in your case you have: #log_e0=ln0=b# where #ln# is the way to indicate the natural log or log in base #e#. But how do you find the right #b# value such that #e^b=0#???? Actually it doesn't work...you cannot find it...you cannot rise to the power of a number and get zero! If you try with a positive #b# it doesn't work (it gets bigger and not zero); for #b=0# it is even worse because you get #e^0=1#! One thing you can do is to manipulate it to get as near as possible to zero... if you take a negative exponent you can get almost there: if #b# is VERY big (negatively) you get very near to zero: for example: #e^-100=1/e^100=3.72xx10^-44#!!!! basically if #b->-oo# then #x=e^b->0# So I would say that #ln0->-oo# using "tends to" instead of “equal to”.

Julia Adams · Answer

undefined The natural logarithm of zero is undefined because there is no number that you can raise to the power of e (approximately 2.71828) to get zero. In mathematical terms, ln(0) is undefined.