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.