# What is the natural log of zero?

Oct 26, 2015

Difficult one!

#### Explanation:

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:
${\log}_{a} x = b \to x = {a}^{b}$
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:
${\log}_{e} 0 = \ln 0 = b$
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: ${e}^{-} 100 = \frac{1}{e} ^ 100 = 3.72 \times {10}^{-} 44$!!!!

basically if $b \to - \infty$ then $x = {e}^{b} \to 0$
So I would say that $\ln 0 \to - \infty$ using "tends to" instead of “equal to”.