i want to talk briefly about how to
visualize power law scaling on a plot
that's done by noticing that there's two
equivalent mathematical expressions
that represent a power law this one the
first one we've seen
y equals c some constant c
times x raised to the alpha power so 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
so those are two equivalent expressions
this one i derived
just by applying the rules of logarithms
that we learned in an earlier unit
if you don't see how i got this just
believe me that that's true
that this is equivalent or go back and
review those rules or
ask about it in the forum if you like
well this is useful for the following
reason
if we plot y equals c times
x raised to the alpha it might look
something like this
for example if alpha is negative
you might notice that if we look at this
equation with the logarithms
it's the equation of a straight line
slope times x plus
some constant if we plot it on what's
called a log log-log plot
in which the y-axis is log y and the
x-axis is log
x so i'm just treating this
as the y-axis and this as the x-axis
and plotting this straight line so the
thing to remember
is that 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
well let's look at a couple of examples
of famous power laws
so the gutenberg richter law of
earthquake magnitudes
you've probably run into this you've
heard of magnitude 1 2
3 etc earthquakes well
magnitude is just the logarithm of the
energy of the earthquake
and what this means is that this is
really a log
scale where we're plotting the log of
the energy
so a magnitude 2 earthquake has
10 times more energy than a magnitude 1
earthquake
magnitude 3 earthquake has 100 times
more energy and so on so
this is a logarithmic scale and here on
this
axis is the number of earthquakes per
year
of a given magnitude it's a straight
line this is a logarithmic
scale also and this is a power law
because it's a straight line on a log
log plot these dots are the actual data
the line is the best fit of a line to
that data it fits it very well
and what it shows is that there's many
somewhere between a hundred and a
thousand magnitude one earthquakes per
year
but very few say
magnitude five or six these only happen
every
so many years and the other earthquakes
of higher magnitude are even rarer so
that's 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
so if you plot the body mass of an
animal on this plot
on this axis versus the metabolic rate
that is the amount of energy in the
metabolism and measured in watts
of animals of different mass and you can
see these different animals it falls on
this straight line
this has actually some very surprising
properties
that we'll be talking about in more
detail in a little while
and 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's
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
and this is city population divided by
the average of all city populations
and this is um the log of that and the
log of this
and you can see that they all scale in a
very similar way
a power law fashion and that actually is
very surprising
and the explanation for it isn't yet
well understood
so you'll hear a little bit about that
later on