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Logarithmic Scales

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Source: Moore's Law, Creative Commons: http://en.wikipedia.org/w/index.php?title=File:Transistor_Count_and_Moore%27s_Law_-_2011.svg&page=1 Other tables and graphs created by the author

In this tutorial, you're going to learn about logarithmic scales. Logarithmic scales are used to adjust the axes on a plot to make it easier to interpret. So let's take a look at this example.

So here's a list of earthquakes based on where and when they occurred, but more importantly, the energy that they release in petajoules, which is millions of millions of joules and their values on the Richter scale. So this row here was the 2004 Christmas Day tsunami that struck Indonesia. And up here, for instance, is the famous San Francisco earthquake of 1906.

When we look at the data plotted on a scatter plot here, what we see is that the data actually follow a curve. As the Richter value goes up, so does the energy released from the earthquake. But it's a curve, it's not a straight line.

The problem with looking at the values this way is-- let's look at this. For instance, the Oaxaca earthquake released 16 petajoules of energy, which is a lot. But compare that to the next one over, the San Francisco earthquake, which released almost four times as much energy, and it looks very, very similar on this particular scale.

So it's hard to interpret exactly these differences here. These are large differences between these two. But it's hard to tell because the values differ by such a large order of magnitude.

So let's take a look. Often, to see those differences, it's beneficial, instead of a standard scale, to use a logarithmic scale. A standard scale uses equal increments. So this distance represents 1,000, no matter how many increments you use. So 1,000, 2,000, 3,000, 4,000, et cetera.

Whereas, a logarithmic scale will use equal ratios. So from here to here might be 10. And then times 10 again. And then times 10 again, times 10 again. So you get to 10,000 here. As opposed to 10 being way way, way down here near the x-axis, it's actually up here. So you can see those differences more clearly.

So for instance, you have something like this on the standard scale for our earthquake data. When you look at it on a logarithmic scale, it looks like this. So this is energy in 10 to the 16th power, 10 to the 17th power, et cetera. Data that appear exponentially curved on a standard scale will often end up appearing linear on a logarithmic scale. And this is an important piece to consider.

One big example of a logarithmic scale being useful is something called Moore's Law. And the law is named after the co-founder of the company Intel, Gordon E. Moore. And he described a trend talking about components on circuits. And he said that the number of components in circuits has doubled every year since the invention of the circuit, and the trend would continue for at least 10 years.

Well, if you look at what actually ended up happening, it actually continued much, much further into the future than he had even predicted. It does appear that between every one and two years, typically around every year, the number of microprocessor transistors has in fact doubled. And this is viewed on a logarithmic scale. And that's why it looks linear. Doubling every year is actually an exponential growth curve.

But here, it looks more linear. So examples of employing a logarithmic scale to show points lying on a line instead of an exponential curve are examples like the Richter scale, examples like Moore's law, but also examples like the decibel scale, which measures sound, and the pH scale, which measures acidity.

So to recap, logarithmic scales are used to show data points that are very far apart. Typically, a rule of thumb is by factors of 100 or more. So if something was 1,000 versus a value of 10, you might want to use a logarithmic scale. Often, data points that show an exponential curve on a standard scale will form a line on a logarithmic scale. And so, we saw that with both the earthquake data and the Moore's law-- the graph that we showed.

So, we talked about logarithmic scales and used Moore's law as an example. Good luck. And we'll see you next time.