I want to prove the natural log can't equal 0. Is this done via the limit definition or the fact the inverse of ln is e and e can't be 0?

How would this proof be started as well thanks.

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- Dec 3rd 2010, 06:37 PMdwsmithProve the natural log can't equal 0
I want to prove the natural log can't equal 0. Is this done via the limit definition or the fact the inverse of ln is e and e can't be 0?

How would this proof be started as well thanks. - Dec 3rd 2010, 06:56 PMDrexel28
- Dec 3rd 2010, 06:56 PMdwsmith
- Dec 3rd 2010, 06:57 PMDrexel28
- Dec 3rd 2010, 06:59 PMdwsmith
I want to show that we can't take the natural log of 0. I wasn't sure if using the inverse function would be how to do it or not.

- Dec 3rd 2010, 07:03 PMDrexel28
I mean, this is all kind of semantical. I could call if I wanted to. If you're asking though, given the predefined function then the inverse function has domain equal to the codomain of , namely and since , we have that isn't (at least not with keeping it as an inverse of the exponential function) defined at zero.

- Dec 3rd 2010, 07:05 PMdwsmith
I don't necessarily want to use e. I was just throwing that out there. I want to prove that 0 isn't in the domain of ln without just saying. Hey the domain is (0, infinity).

- Dec 3rd 2010, 07:09 PMDrexel28
But that doesn't make sense. If you don't define as being the inverse of then I'd ask you 'what is it' and no matter how you've answered (assuming you've done so rigorously) you will, by necessity, have answered your own question. You can't define without specifying its domain. A function can be thought of as the ordered triple where are sets and is a relation such that for each there is precisely one for which .

- Dec 3rd 2010, 07:24 PMdwsmith
I found a method using the derivative of the ln

From this equation, we can easily see that and - Dec 3rd 2010, 07:30 PMDrexel28
- Dec 3rd 2010, 08:49 PMCaptainBlack
When I see something like this it convinces me that indefinite integration should never be taught.

That does not define the natural logarithm. It leaves it defined up to an additive constant. You need to define it as a definite integral when the non-convergence of that integral when the upper limit is zero shows that the natural log of zero does not exist.

CB - Dec 3rd 2010, 10:55 PMchisigma
- Dec 3rd 2010, 11:23 PMCaptainBlack
- Dec 4th 2010, 03:49 AMtonio

Take the very basic, high-schoolish, definition of logarithm: , when we already

know-define-demmand that . From this, for some

which, as Drexel already pointed out, is impossible by the very

definition of power of positive real numbers (although he talked about inverse function and

stuff, which isn't necessary in a first approach to this)

Tonio - Dec 4th 2010, 04:05 AMHallsofIvy
In other words it makes no sense to try to proved

**anything**about a function without saying how you have**defined**it! One way to show that 0 is not in the domain of ln(x) is to define ln(x) is as the inverse function of and then show that is not true for any x. How you do that would, of course, depend upon how you defined .

Another way to define ln(x) is

You can then show that 0 is not in the domain of ln(x) by showing that does not exist.