Monday Math: Logarithms

We begin with a joke. What’s a logarithm? It’s a birth control method for lumberjacks. Hahahahaha! Believe it or not, one of my high school math teachers taught me that.

Actually, logarithms are a computational tool for turning products into sums. They are defined as follows.


\[
\log_a b=x \textrm{ if and only if } a^x=b.
\]

 

The thing on the left is read, “Log to the base a of b.” It can be thought of as the power to which a must be raised to obtain b

Two simple examples are


\[
\log_2 32 =5 \phantom{xxx} \textrm{and} \phantom{xxx} \log_7 49=2.
\]

 

What do we mean when we say that logarithms turn products into sums?

Well, suppose that


\[
\log_a x=s \phantom{xxx} \textrm{and} \phantom{xxx} \log_a y =t
\]

 

That implies that


\[
a^s=x \phantom{xxx} \textrm{and} \phantom{xxx} a^t=y.
\]

 

But, recalling the basic rules of exponents, we now have


\[
\log_a (xy) = \log_a (a^s a^t) = \log_a (a^{s+t})=s+t=\log_a x + \log_a y
\]

 

Like I said, it turns products into sums.

One of the more endearing thing about logarithms is the strange terminology that gets associated with them. For example, base ten logs are usually referred to as “common logarithms” even though mathematicians rarely use them. On the other hand, logs to the base e are said to be “natural logarithms” even though e is a weird, irrational number that does not seem natural at all.

Which raises another question: Whose bright idea was it to use the notation ln for the natural logarithm? I mean, really, have you noticed that it is completely unpronounceable? When I was in high school I learned that this was pronounced “lawn,” like the thing you mow. When I got to college I got laughed at — laughed at! — for saying that. Apparently everyone else was learning to pronounce it “lynn,” like a woman’s name. That’s not very good of course, since it sounds too much like “lim,” meaning limit. Some people would say, “ell en,” but if we have been reduced to spelling things out then truly all hope is lost.

The fact is that mathematicians, at least when doing research, nearly always just write log without specifying a base. It is simply assumed that natural logarithms are intended.

 

So what is so natural about logs to the base e?

For one thing, the natural logarithm function has a very nice definition in terms of areas. We have the equation:


\[
\ln x = \int_1^x \frac{1}{t} \ dt
\]

 

If you are unfamiliar with calculus, the expression on the right refers to the area underneath the curve 1/x between one and x.

Even cooler is the fact that


\[
f(x)=e^x
\]

 

is the only function (up to multiplication by a constant) that is equal to its own derivative. If that seem marvelously coincidental, it isn’t. The number e is specifically defined to make that true. If you remember your calculus, then you know that we determine the derivative of an arbitrary exponential function as follows:


\[
f'(x)=\lim_{h \rightarrow 0} \frac{f(x+h)-f(x)}{h}=\lim_{h \rightarrow 0} \frac{a^{x+h}-a^x}{h}
\]

 

If we now employ the laws of exponents, factor the top, and recall that we can factor constants out of limits, we obtain


\[
f'(x)=a^x \lim_{h \rightarrow 0} \left( \frac{a^h-1}{h} \right)
\]

 

But what is the value of that limit in parentheses? It will depend on a, of course. It can be shown using calculus that there is a unique value of a for which the limit is equal to one. That number is defined to be e.

 

Let me remind you that the goal of this whole series has been to prove that the sum of the reciprocals of the primes diverges. We have made great progress in that direction. Last week we established that the harmonic series, which we know to diverge, can be expressed as an infinite product indexed over the primes. Now we can take the logarithm of that product to obtain a sum indexed over the primes. We need just one more ingredient. Taylor series! Stay tuned…

Comments

  1. #1 Eric Juve
    August 23, 2010

    ln is “lawn” as long as I have been aware of it.

  2. #2 Pierce R. Butler
    August 23, 2010

    “Get off my ln!” just doesn’t have the right ring to it.

  3. #3 Greg Laden
    August 23, 2010

    That’s some kick-ass math typesetting you’ve got going there.

  4. #4 The Science Pundit
    August 23, 2010

    Maybe it’s because I was also taught that lb was the abreviation of pound, but I always pronounced it “log” even when I wrote it as ln.

  5. #5 Hamilton Jacobi
    August 23, 2010

    I also pronounce “ln” as “log”. Not sure where I learned it.

  6. #6 Benton Jackson
    August 23, 2010

    When the waters of the Flood subsided, Noah spake unto the animals: “Go forth and multiply.” All the animals did as they were bid, save a pair of serpents, who said, “We cannot multiply, for we are adders.” Then Noah bade them, “Come thou hither, upon the rough furniture where I partake my meals.” They did so, and in due time, became exceeding numerous, for even adders can multiply with a log table.

  7. #7 Jason Rosenhouse
    August 23, 2010

    Benton –

    That’s even worse than the joke with which I started the post! :)

    Greg –

    Go here if you want to see how the magic is done.

  8. #8 Sam C
    August 24, 2010

    Interesting to see your take on logarithms in the same way as any other interesting mathematical function.

    For us older folk, who had to do calculations before electronic calculators, logs have a different place in our lives because we not only could use them to convert products into sums, we actually did use them, with our 4 or 7 digit log tables, using bar numbers for numbers less than one.

    Slide rules (also based on logs) were good for quick and dirty calcs, but you needed log tables for a more precise answer. Hence the effort put into calculating accurate tables for navigators.
    As calculators have made multiplication as easy as addition, common (base 10) logs have lost their main utility, so natural logs can be used almost always.

  9. #9 Raskolnikov
    August 24, 2010

    But “ell en” is like a French woman’s name! (Hélène)

  10. #10 KeithB
    August 24, 2010

    Don’t be dissin’ common logs – we engineers use them all the time. Especially when working with dB (decibels).

  11. #11 heddle
    August 24, 2010

    Jason,

    The fact is that mathematicians, at least when doing research, nearly always just write log without specifying a base. It is simply assumed that natural logarithms are intended.

    No surprise there. You’se people also mix up the polar and azimuthal angles to. (It’s θ and φ, not the other way around. Apostates.)

  12. #12 Alejandro
    August 24, 2010

    Favorite log joke: ask someone who knows calculus:

    “What is the integral of 1/cabin d cabin?”
    “Er… Aha! A log cabin!”
    “No, a houseboat! You forgot to add the C!”

  13. #13 James Sweet
    August 25, 2010

    Of course then you have your computer scientists. When they say “log”, they obviously mean base 2. (Except when they don’t…. remember, as far as a computer scientist is concerned, there’s no difference between 2*x and 2000000*x. They are both linear time!)

  14. #14 Motena
    October 16, 2010

    I never really understood a thing about logarithms
    in class,well now at least now i know. thanx!

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