# ln 0 is Undefined

Here is a simple proof that the natural log of zero or (ln 0) is undefined.

First, assume:

If you apply Euler's number (e) to both sides of the equation, this would be the result:

Since e ^ (ln y) = y, we can substitute to get this:

For all values of x, e ^ x will always be a positive number and never be equal to zero. Only if x could be equal to negative infinity, would e ^x approach zero. However, since negative infinity is not a number, that is not a valid substitution.

Therefore, since there is no value of x that could make this equation (0 = e ^ x) true, then (ln 0) is undefined.

by Phil for Humanity

on 09/13/2011