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ln 0 is Undefined


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

First, assume:

ln 0 = x


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

e ^ (ln 0) = e ^ x


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

0 = e ^ x


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

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