Just to be clear, 10^{6} means 10 x 10 x 10 x 10 x 10 x 10. But what if I want to do the inverse of 10 raised to some power? It’s much easier to write big numbers by raising them to some power—this is exactly what we do with numbers in scientific notation. Finding the power of 10 that a number is raised to is exactly what a logarithm does. If I take the log of 1,000,000, it gives the result of 6. Oh, here is an important note. If we are talking about 10 raised to some power, that means we are using a log base of 10. The two most common bases are 10 (because we write numbers in base-10) or e, the natural number where e is approximately 2.718 (it’s irrational). Here is a more detailed explanation of e.

But wait! You can also take the logarithm for numbers that aren’t integer powers of 10. Let’s just pick a number—I’m going with 1,234. If I take the logarithm of this number, I get:

This means that if you raise 10 to the power of 3.09132, you get 1,234. But why? Why would you do that? OK, let’s go back to our terrible Covid data. Suppose that instead of plotting the number of confirmed infections, I plot the log (base 10) of the number of infections. I can then plot the log of the number vs. the day number. Here’s what that looks like.

Just to be clear—this is the same data as the first plot, but there is a big difference. You can actually see the data for South Korea even though that country’s numbers are so much lower than those in the USA. Why? Well, let’s look at the total number of confirmed cases as of November 17, 2020. For the USA, it’s 11,036,935 and for South Korea it’s 28,769. Now let’s take the log (base 10) of both of these numbers.

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