What’s the Deal with Logarithms?

I’m going to talk about logs here. I have more to say later, but this is a basic intro sketch.

First I’m going to talk about the stuff of elementary school. When it comes to mathematics, most people find comfort in elementary school mathematics.

So, consider the humble number line:

We want to move along it. What can we do? We could:

1. Start at one place and move a certain size step to the right. This is called addition.
2. Start at one place and move a certain size step to the left. This is called subtraction.
3. Start in the middle (which we declare to be 0). We’ll decide on our step size, and then we’ll move that step size a certain number of times. This is called multiplication.
4. Start in the middle and decide where we want to end. We’ll decide on the size of the steps we’re going to take, and then figure out how many steps we need to take. This is called division.

For instance:

1. 3 + 2 says, “Start at 3 and move a step of size 2 to the right. Where do you end?”
2. 5 − 3 says, “Start at 5 and move a step of size 3 to the left. Where do you end?”
3. 6 × 3 says, “Start at 0. Taking steps of size 6 to the right, move 3 times. Where do you end?”
4. 25 ÷ 5 says, “Start at 0 and end at 25. Each step will be size 5. How many steps do you take?”

We could wonder what happens when we go to the left of 0. We could also wonder what it means to take 4.5 steps, or what it means to take π steps to get to e, or a bunch of other questions. That involves extending our statements above in reasonable, fairly predictable ways.

I’m going to talk about logarithms now.

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Once upon a time, mathematicians and engineers used to use something called a slide rule. The handheld calculator made the slide rule obsolete, but it did have a feature that’s important to my discussion.

Look specifically at the rows marked A and B. Look at how far the 1 and 2 are, while the 2 and 3 are closer, the 3 and 4 are closer still, and so on. In fact, the 1 and 2 are as far apart as the 2 and 4, which are as far apart as the 4 and 8. Also, the 1 and 3 are as far apart as the 3 and 9.

The A and B rows on a standard slide rule use a logarithmic scale. The logarithmic scale has the property that the numbers are marked off such that they’re exponentially distributed.

For instance, here’s a log scale with the powers of 2 highlighted. Notice how they’re spread out.

Here’s the same log scale, but with the powers of 3 highlighted.

And here it is with the powers of 5 highlighted.

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We can’t use slide rules to add or subtract, though. Let’s look at what we can do with the logarithmic scale.

We want to move along it. What can we do? We could:

1. Start at one place and move a certain size step to the right. This is called multiplication.
2. Start at one place and move a certain size step to the left. This is called division.
3. Start in the middle (which we declare to be 1). We’ll decide on our step size, and then we’ll move that step size a certain number of times. This is called exponentiation.
4. Start in the middle and decide where we want to end. We’ll decide on the size of the steps we’re going to take, and then figure out how many steps we need to take. This is called finding the logarithm.

For instance:

1. 3 × 2 says, “Start at 3 and move a step of size 2 to the right. Where do you end?”
2. 6 ÷ 3 says, “Start at 6 and move a step of size 3 to the left. Where do you end?”
3. 6³ says, “Start at 1. Taking steps of size 6 to the right, move 3 times. Where do you end?”
4. log₆ 216 says, “Start at 1 and end at 216. Each step will be size 6. How many steps do you take?”

Notice that these are the same exact ways of moving as our friends from elementary school, we’re just moving along a different scale. Instead of the numbers themselves being spread out evenly, it’s the logarithms that are spread out evenly.

Indeed, the basis of this entire system is the logarithm, since that determines the step size for moving around. A step of size 4 is twice as much as a step of size 2. A step of size 27 is three times as much as a step of size 3.

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“Wait… I thought division was the other way around.”

On the number line, there are two ways to interpret division:

1. Start in the middle and decide where we want to end. We’ll decide on the size of the steps we’re going to take, and then figure out how many steps we need to take. This is called division.
2. Start in the middle and decide where we want to end. We’ll decide on the number of steps we’re going to take, and then figure out the size we need. This is also called division.

Only the first one matches up with logarithms. This is because multiplication is commutative, meaning that it doesn’t matter which order it’s written in (e.g., 5 × 7 = 7 × 5), but exponentiation isn’t (e.g., 5⁷ doesn’t equal 7⁵).

The second one does match up with a thing we do: Radicals. For instance, ∛216 says, “Start at 1 and end at 216. Take 3 steps to get there. How big will each step be?”

So, using the number line, division can tell us EITHER the number of steps needed OR the size of the steps, but logarithms can only tell us the number of steps, while radicals can only tell us the size of the steps.

We could then ask: What if we want to take half a step of size 2 on the number line? (We get to 1.414…, that is, √2.) What if we want to take 4.5 steps of size 2.5? The instructions get complicated, but you could use the log scale to find that 2.5^4.5 = 61.76…. What if we want to take steps of size 5.2 to get to 8.5? Well, log_(5.2) 8.5 = 1.30….

Note that the numbers do get messier than with the linear scale, but the process is the same. In a future article, I’ll give more specifics.

I’ll end with this, one of my favorite YouTube videos. A Vi Hart classic.