The Logarithmic Property

Whoops.. Between preparing my exam, practicing my rumba, and adapting to the new WordPress interface, I forgot to actually post today’s installment :-/

Yesterday we defined the natural logarithm as the function

$\displaystyle\ln(x)=\int\limits_1^x\frac{dt}{t}$

on the interval $\left(0,\infty\right)$ . This function is differentiable everywhere in this interval, and its derivative is $\frac{1}{x}$ at each point $x$ .

We call this function a logarithm because it satisfies the “logarithmic property”. Simply put, it’s a homomorphism of groups from the group of positive real numbers under multiplication to the group of all real numbers under addition.

That is, since the real numbers are an ordered field they are a fortiori a group if we just throw away the multiplication and order structures. Also, if we get rid of that pesky noninvertible ${0}$ element, they’re a group under multiplication, and the positive elements are a subgroup. The logarithm takes elements of this group and sends them to the additive group, and the homomorphism property reads: $f(xy)=f(x)+f(y)$ . In particular, we must have $f(1)=0$ .

So is our “natural logarithm” a logarithm? First off, we can easily check that

$\displaystyle\ln(1)=\int\limits_1^1\frac{dt}{t}=0$

As for the other property, let’s write

$\displaystyle\ln(xy)=\int\limits_1^{xy}\frac{dt}{t}=\int\limits_1^x\frac{dt}{t}+\int\limits_x^{xy}\frac{dt}{t}=\ln(x)+\int\limits_x^{xy}\frac{dt}{t}$

Now let’s take the second term on the right here and perform a change of variables, setting $u=\frac{t}{x}$ . Then we have $du=\frac{dt}{x}$ , and as $t$ runs over $\left[x,xy\right]$ the new variable $u$ runs over $\left[1,y\right]$ . That is, we have

$\displaystyle\int\limits_x^{xy}\frac{dt}{t}=\int\limits_1^y\frac{du}{u}=\ln(y)$

and the logarithmic property holds.

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April 9, 2008 - Posted by John Armstrong | Analysis, Calculus

9 Comments »

And, since this homomorphism is actually an isomorphism (if I am correct), we could posit the existence of an inverse logarithmic function based on what we already know!

Comment by Vishal Lama | April 9, 2008 | Reply
Give it until this afternoon. Since I’m posting every day I try not to make them too huge.

Comment by John Armstrong | April 9, 2008 | Reply
Are you taking dancing lessons?

Comment by Michael | April 9, 2008 | Reply
Michael: yes, I am, through Tulane’s ballroom club. Even if I’m the old guy around there and I’m not about to meet anyone to socialize with aside from the lessons and practices, it’s good to have some sort of hobby that leads to actual human contact. Writing this stuff and watching old/arty movies on cable gives one a bit of the cabin fever after a while.

By the way, I’m sorry that your comments keep getting caught up in Akismet’s filter. Whenever I go through and find them I tell it not to hate on you, but to no avail.

Comment by John Armstrong | April 10, 2008 | Reply
My wife and I took dancing lessons at my health club several years ago. We loved it. We stopped because at the next level they wanted us to take private lesson and that didn’t seem as much fun. It wasn’t that we wanted to hide in crowd, but as you said the social aspect was nice.

And learning how to move gracefully listening to good music [they played a lot of swing] is nice. I keep thinking of doing it again; I like dancing.

BTW we were the oldest in the class by at least 15 years.

Comment by Michael | April 10, 2008 | Reply
You misspelled ‘a fortiori’. This is interesting to me because it seems your misspelling was influenced by the spelling of words like interior.

The rest of the post was interesting.

Comment by Robert | April 15, 2008 | Reply
You’re right, Robert, though the spelling I used is a common enough mistake that my real-time spell-checker didn’t catch it. Thanks.

Comment by John Armstrong | April 15, 2008 | Reply
[...] Exponential Property We’ve defined the natural logarithm and shown that it is, in fact, a logarithm. That is, it’s a homomorphism from the multiplicative group [...]

Pingback by The Exponential Property « The Unapologetic Mathematician | April 16, 2008 | Reply
Michael: I actually wouldn’t mind a private lesson with a fixed partner. Yeah, socializing in a group is one thing, but when I say human contact I mean with anyone. If I had a wife as company I might not mind staying at home so much.

Comment by John Armstrong | April 19, 2008 | Reply

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