Wednesday, August 5, 2009

More about e

The exponential function comes up all the time in science. Often the rate of change of some quantity x is proportional to the value x itself:

dx/dt = k * x


where k is a constant. Everyday examples for biology include bacterial growth and radioactive decay. As most of you know, we can solve this problem using a fundamental result from calculus:

d/dx ln(x) = 1/x
ln(x) = ∫ 1/x dx


So, rearranging the first equation and then integrating:

dx/x = k * dt
∫ dx/x = ∫ k * dt
ln(x) = k * t
x = A exp { k * t }


Evaluate the constant A by setting t = 0, so then x = x0:

x = x0 exp { k * t }


At a special value, t = T (one generation, one half-life):

x = 2 * x0
2 = exp { k * t }
ln(2) = k * T


I remember reading something by Francois Jacob, a long time ago, in which he described his and Jacques Monod's excitement at some point that they had done a set of experiments and the value 0.69 appeared from their calculations for all the experiments. What was this magic number? It took a minute before the penny dropped, and they remembered that ln(2) = 0.693:

>>> import math
>>> print round(log(2),3)
0.693


I haven't been able to find the reference.

After that long preamble, here is the real subject of this post. Starting from:

(1)  d/dx ex = ex


we want to show that:

d/dx ln(x) = 1/x


By definition:

     eln(x) = x


Take the derivative on both sides:

     d/dx eln(x) = d/dx x


The right side equals 1. That is:

(2)  d/dx eln(x) = 1


Now for the left side. It is easier if we make a substitution. Let:

     u = ln(x)
eln(x) = eu = x


And

     d/dx eln(x) = d/dx eu


We use the chain rule:

     d/dx eu = d/du eu * du/dx


By the definition of e above in (1):

     d/du eu = eu


So:

     d/dx eu = eu * du/dx


Here is the trick. Substitute u = ln(x) to go back to x:

     d/dx eu    = x * du/dx
d/dx eln(x) = x * du/dx
d/dx eln(x) = x * d/dx ln(x)


But from (2), the left side equals 1:

     1 = x * d/dx ln(x)
d/dx ln(x) = 1/x


Or, quoting Heath's translation of Euclid: "precisely what was required to be proved."

I would love to credit the source for this, but I've forgotten where I found it.

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