8.6 Logarithm concepts -- graphing logarithmic functions

Suppose you want to graph the function $y={\text{log}}_2(x)$ . You might start by making a table that looks something like this:

This doesn’t seem to be the right strategy. Many of those numbers are just too hard to work with.

So, you start looking for numbers that are easy to work with. And you remember that it’s important to look at numbers that are less than 1, as well as greater. And eventually, you end up with something more like this.

In fact, the easiest way to generate this table is to recognize that it is the table of $y=2^x$ values, only with the $x$ and $y$ coordinates switched! In other words, we have re-discovered what we already knew: that $y=2^x$ and $y={\text{log}}_2(x)$ are inverse functions.

When you graph it, you end up with something like this:

As always, you can learn a great deal about the log function by reading the graph.

• The domain is $x>0$ . (You can’t take the log of 0 or a negative number—do you remember why?).
• The range, on the other hand, is all numbers. Of course, all this inverses the function $2^x$ , which has a domain of all numbers and a range of $y>0$ .
• As $x$ gets closer and closer to 0, the function dives down to smaller and smaller negative numbers. So the $y$ -axis serves as an “asymptote” for the graph, meaning a line that the graph approaches closer and closer to without ever touching.
• As $x$ moves to the right, the graph grows—but more and more slowly. As $x$ goes from 4 to 8, the graph goes up by 1. As $x$ goes from 8 to 16, the graph goes up by another 1. It doesn’t make it up another 1 until $x$ reaches 32...and so on.

This pattern of slower and slower growth is one of the most important characteristics of the log. It can be used to “slow down” functions that have too wide a range to be practical to work with.

Using the log to model a real world problem

Lewis Fry Richardson (1881–1953) was a British meteorologist and mathematician. He was also an active Quaker and committed pacifist, and was one of the first men to apply statistics to the study of human conflict. Richardson catalogued 315 wars between 1820 and 1950, and categorized them by how many deaths they caused. At one end of the scale is a deadly quarrel, which might result in 1 or 2 deaths. At the other extreme are World War I and World War II, which are responsible for roughly 10 million deaths each.

As you can see from the chart above, working with these numbers is extremely difficult: on a scale from 0 to 10 Million, there is no visible difference between (say) 1 and 100,000. Richardson solved this problem by taking the common log of the number of deaths . So a conflict with 1,000 deaths is given a magnitude of $\text{log}(\text{1000})=3$ . On this scale, which is now the standard for conflict measurement, the magnitudes of all wars can be easily represented.

Richardson’s scale makes it practical to chart, discuss, and compare wars and battles from the smallest to the biggest. For instance, he discovered that each time you move up by one on the scale—that is, each time the number of deaths multiplies by 10—the number of conflicts drops in a third. (So there are roughly three times as many “magnitude 5” wars as “magnitude 6,” and so on.)

The log is useful here because the logarithm function itself grows so slowly that it compresses the entire 1-to-10,000,000 range into a 0-to-7 scale. As you will see in the text, the same trick is used—for the same reason—in fields ranging from earthquakes to sound waves.