Powers and exponentials

To understand how one variable depends on another, we introduced the idea of functional dependence.  This helps us understand not just that one variable depends on another, but how it depends on the other. This is especially important in complex situations such as biology where many variables can be involved and "which one dominates" matters.

Powers

Some of the most useful and convenient functional dependencies that we will encounter are power laws. The variable we choose to be dependent depends on the variable we choose to be independent by some power of that independent variable. Thus

$y = f (x ) = x^N  = x$ multiplied by itself N times

says that "y goes like x raised to the N^th power ".

Exponentials

We know that a quadratic function rises faster than a linear one (eventually) and a cubic rises faster than a quadratic. But there is an extremely useful function that eventually rises faster than any power. This is the exponential function. In this case, the variable is not raised to a power — it is in the power that some constant is raised to. So as the variable gets bigger and bigger, the power the constant is raised to gets bigger and bigger.

One special number that appears often is the constant called "e".

We write

$$y = e^x$$ .

Now e could be any constant (and is often 10 or 2), but we typically take it as a special transcendental number (that means it's decimal representation never stops and never repeats): e = 2.712...  We make this choice because e is the constant for which the function $e^x$ is its own derivative.  That is, if we write $y=e^x$  then

          $$\frac{dy}{dx} = y$$

Note that a power law should not be referred to as an "exponential dependence" even though the variable "has an exponent". That terminology is reserved for the case when the variable is in the exponent.

In case you wondered how in the world one knows what "e" to the something is, the following result allows you to calculate (eventually) the value of e raised to some number.

$$e^x = 1 + x + \frac{x^2}{1*2} + \frac{x^3}{1*2*3} + \frac{x^4}{1*2*3*4} + ... $$

You can see how this goes on -- and it goes on forever.  But for any fixed value of $x$, the denominators grow faster than powers do. Eventually, those denominators start making the terms smaller and smaller until they are so small that they can be dropped.

While this looks really messy and we're not going to calculate with it (just use it to learn about the properties of the exponential), looking at it gives us three interesting messages.

1. It explains why the exponential function is its own derivative.

If you take the derivative of the power series representation of the exponential, something interesting happens. The first term vanishes, the derivative of the linear term becomes the old first term (1), the derivative of the third (quadratic) term becomes the old second (linear) term, etc.  So the derivative of each term becomes the previous term in the original series.  We wind up getting the same thing back that we started with.  This also shows us why the denominators have the structure they do. [And with a little fancy mathematical footwork, we can show that the exponential is the only function that is its own derivative.]

2. It shows that we can only take exponentials of pure numbers.

Since we know that you can't add a length and an area -- or any dimensioned quantity to its square -- the power series only makes sense if "x"  is a pure number.  You can't take an exponential of a quantity that has units.  Whenever in science we have an exponential, it will always be the ratio of two quantities with the same units -- typically a variable and a scale for that variable.  (Like a time and a rate constant.)

3. It shows why an exponential grows faster than any power.

We know that x² grows faster than x and x³ grows faster than x², etc. Since the exponential contains all powers, whatever power you choose, the exponential has terms that grow faster than that. This is why an exponential is so powerful and saying that something "grows exponentially" is such a strong statement. [And this is why it's important NOT to be sloppy in your language and call a variable raised to a power as "growing exponentially". $x^2$ is quadratic growth, NOT exponential growth.]

Logarithms

The exponential function does the interesting thing of converting sums into products.  If $R = e^a$ and $S = e^b$ then $RS = e^{a+b}$.

So if $f (x) = e^x$ then we have

$$f(x_1)f(x_2) = f(x_1 + x_2)$$

Since multiplying is harder than adding, it's sometimes useful to go backwards from the exponential function. 

Taking the exponential of x and setting $y=e^x$, if we are given $x$ we can use our calculator (or series) to find $y$. But what if we are given $y$ and want to find $x$?  The answer to that is called the natural logarithm of $y$.  That gives us the equation:

$$y=e^{ln(y)}$$.

This shows that the natural log (ln) is the inverse function of the exponential. If we first take the natural log and then exponentiate it, we get back what we started with. It works the other way too:

$$x=ln(e^x)$$

If we exponentiate first and then take the natural log, we get back what we started with. So the natural log function (ln) undoes the exponential function and vice versa.

We will see in the follow-ons that power laws and exponentials  are very useful in modeling physical systems. Logarithms are particularly valuable in analyzing data and seeing whether something behaves like a power law. 

Joe Redish 8/14/11