Posted by 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…
August 23, 2010
August 23, 2010
August 23, 2010
August 23, 2010
August 23, 2010
August 23, 2010
August 23, 2010
August 24, 2010
August 24, 2010
August 24, 2010
August 24, 2010
August 24, 2010
August 25, 2010
October 16, 2010