I want to talk briefly about how to

visualize power law scaling on a plot.

That's done by noticing that there are

two equivalent mathematical expressions

that represent a power law. The first one

we have seen, y equals c, some constant,

times x raised to the power alpha.

You just saw that. If you take the

logarithm of both sides of this equation,

you get log of y equals alpha times

log of x plus log of c. Those are two

equivalent expressions. This one I

derived just by applying the rules of

logarithms we learned in an earlier unit.

If you don't see how I got this, just

believe me that that is true. That this

is equivalent, or go back and review those

rules or ask about it in the forum, if you

like. This is useful for the following

reason. If we plot y = c x ^ alpha, it

might look something like this,

for example, if alpha is negative, you

might notice if we look at this equation

with the logarithms, it's the equation of

a straight line, slope times x + some

constant. If we plot it on what's called

log-log plot, in which the y-axis is

log y and the x axis is log x. I am just

treating this as the y axis and this as

the x axis, and plotting this straight

line. The thing to remember is on a

regular plot, a power law looks something

like this. On a log-log plot, it looks

like a straight line where the slope is

the exponent alpha. Let's look at a

couple of examples, of famous power laws,

The Gutenberg-Richter law of earthquake

magnitudes, you've probably run into this.

You have heard of magnitude 1, 2, and 3

earthquakes. Well, magnitude is just the

logarithm of the energy of the earthquake.

What this means is that this is really a

log scale, where we are plotting the log

of the energy. Magnitude 2 earthquake

has 10 times more energy than a magnitude

one earthquake. Magnitude 3 earthquake

has 100 times more energy and so on.

This is a logarithm scale, and here on

this axis is the number of earthquakes per

year of a given magnitude. It is a

straight line. This is a logarithmic

scale also. This is a power law

because this is a straight line on a

log-log plot. These dots are the actual

data. This line is the best fit to the

data. It is a very good fit. It shows

that there are many, somewhere between

a hundred and thousand magnitude 1

earthquakes per year, but very few, say,

magnitude 5 or 6. These only happen

every few years. Earthquakes of even

higher magnitude are even rarer. This is

a very famous power law. One famous power

law that we're going to be talking about

in this unit is the law of metabolic

scaling in animals. If you plot the body

mass of an animal on this axis vs. the

metabolic rate, that is, the amount of

energy in the metabolism, measured in

Watts, of animals of different mass, you

can see these different animals, it falls

on a straight line. This has some very

surprising properties we'll be talking

about in more detail in a little while.

Finally, you'll hear very briefly about

some relatively new work about scaling in

cities. This plot is from a paper by

Bettencourt and West, that came out in

2010, a link to it is posted on our

course materials page, and it shows how

properties of cities such as crime, gross

domestic product, income and patents,

scale with the population of the city.

This is the city population divided by the

average of all city populations, and this

is the log of that, and the log of this,

and you can see that they all scale in a

very similar way, in a power law fashion.

That is very surprising and the

explanation for it isn't yet

well-understood. You'll hear a little

bit about that later on.