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.
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.
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 .
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
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
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.