Prove the natural log can't equal 0

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December 3rd 2010, 06:37 PM
dwsmith

Prove 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.
December 3rd 2010, 06:56 PM
Drexel28

Quote:

Originally Posted by dwsmith

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 don't understand. $\ln(1)=0$ ?
December 3rd 2010, 06:56 PM
dwsmith

$\displaystyle ln(0)=DNE$
December 3rd 2010, 06:57 PM
Drexel28

Quote:

Originally Posted by dwsmith

$\displaystyle ln(0)=DNE$

So the question isn't 'why can't the natural logarithm ever equal zero' as you stated, but instead 'why can't $e^x$ ever equal zero'?
December 3rd 2010, 06:59 PM
dwsmith

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.
December 3rd 2010, 07:03 PM
Drexel28

Quote:

Originally Posted by dwsmith

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.

I mean, this is all kind of semantical. I could call $\ln(0)=e$ if I wanted to. If you're asking though, given the predefined function $\exp:\mathbb{R}\to\mathbb{R}^+$ then the inverse function $\exp^{-1}$ has domain equal to the codomain of $\exp$ , namely $\mathbb{R}^+$ and since $0\notin\mathbb{R}^+$ , we have that $\exp^{-1}$ isn't (at least not with keeping it as an inverse of the exponential function) defined at zero.
December 3rd 2010, 07:05 PM
dwsmith

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).
December 3rd 2010, 07:09 PM
Drexel28

Quote:

Originally Posted by dwsmith

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).

But that doesn't make sense. If you don't define $\ln$ as being the inverse of $\exp$ 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 $\ln$ without specifying its domain. A function can be thought of as the ordered triple $\left(A,B,f\right)$ where $A,B$ are sets and $f\subseteq A\times B$ is a relation such that for each $a\in A$ there is precisely one $b\in B$ for which $(a,b)\in f$ .
December 3rd 2010, 07:24 PM
dwsmith

I found a method using the derivative of the ln

$\displaystyle ln(|t|)=\int\frac{dt}{t}$

From this equation, we can easily see that $\displaystyle t\neq 0$ and $\displaystyle |t|\in\mathbb{R}^+$
December 3rd 2010, 07:30 PM
Drexel28

Quote:

Originally Posted by dwsmith

I found a method using the derivative of the ln

$\displaystyle ln(|t|)=\int\frac{dt}{t}$

From this equation, we can easily see that $\displaystyle t\neq 0$ and $\displaystyle |t|\in\mathbb{R}^+$

But that isn't really well-defined. Really what you may want to consider is that $\displaystyle \ln(x)=\int_1^x \frac{dt}{t}$ and that if you take the domain to the largest for which that equation makes sense then surely $0$ isn't included. But that's really quite silly.
December 3rd 2010, 08:49 PM
CaptainBlack

Quote:

Originally Posted by dwsmith

I found a method using the derivative of the ln

$\displaystyle ln(|t|)=\int\frac{dt}{t}$

From this equation, we can easily see that $\displaystyle t\neq 0$ and $\displaystyle |t|\in\mathbb{R}^+$

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
December 3rd 2010, 10:55 PM
chisigma

Quote:

Originally Posted by dwsmith

I found a method using the derivative of the ln

$\displaystyle ln(|t|)=\int\frac{dt}{t}$

From this equation, we can easily see that $\displaystyle t\neq 0$ and $\displaystyle |t|\in\mathbb{R}^+$

That is more exact to say that...

$\displaystyle \ln |t| = \left\{\begin{array}{ll}\int_{-1}^{t} \frac{d \tau}{\tau} ,\,\,t<0\\{}\\ \int_{1}^{t} \frac{d \tau}{\tau} ,\,\, t > 0\end{array}\right.$

In any case the function $\ln |t|$ has a singularity in $t=0$ ...

Kind regards

$\chi$ $\sigma$
December 3rd 2010, 11:23 PM
CaptainBlack

Quote:

Originally Posted by chisigma

That is more exact to say that...

$\displaystyle \ln |t| = \left\{\begin{array}{ll}\int_{-1}^{t} \frac{d \tau}{\tau} ,\,\,t<0\\{}\\ \int_{1}^{t} \frac{d \tau}{\tau} ,\,\, t > 0\end{array}\right.$

In any case the function $\ln |t|$ has a singularity in $t=0$ ...

Kind regards

$\chi$ $\sigma$

Then:

$\displaystyle \ln(0.5)=\ln(|-0.5|)=\int_{-1}^{-0.5} \dfrac{1}{\tau}\ d\tau<0\ ?$

CB
December 4th 2010, 03:49 AM
tonio

Quote:

Originally Posted by dwsmith

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.

Take the very basic, high-schoolish, definition of logarithm: $\log_ab=x\Longleftrightarrow a^x=b$ , when we already

know-define-demmand that $a,b>0\,,\,a\neq 1$ . From this, $b=0\Longrightarrow a^x=0$ for some

$x\in\mathbb{R}$ 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
December 4th 2010, 04:05 AM
HallsofIvy

Quote:

Originally Posted by Drexel28

I mean, this is all kind of semantical. I could call $\ln(0)=e$ if I wanted to. If you're asking though, given the predefined function $\exp:\mathbb{R}\to\mathbb{R}^+$ then the inverse function $\exp^{-1}$ has domain equal to the codomain of $\exp$ , namely $\mathbb{R}^+$ and since $0\notin\mathbb{R}^+$ , we have that $\exp^{-1}$ isn't (at least not with keeping it as an inverse of the exponential function) defined at zero.

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 $f(x)= e^x$ and then show that $e^x= 0$ is not true for any x. How you do that would, of course, depend upon how you defined $e^x$ .

Another way to define ln(x) is
$ln(x)= \int_1^x \frac{1}{t}dt$

You can then show that 0 is not in the domain of ln(x) by showing that $\int_1^0 \frac{1}{t}dt$ does not exist.