Mathematical Journeys: Undoing the Unknown Exponent

This journey is heavily inspired by the youtube mathematician Vi Hart, whose videos describing mathematical concepts through doodling in a notebook were the inspiration for much of my mathematical journeys series. I'll put a link to her video on this topic at the end of the journey, and I highly encourage everyone to go check her out.

Let's talk exponents.

But to do that, first we should talk about multiplication. Multiplication is a shortcut for adding a bunch of the same number together. If I gave you:

5 + 5 + 5 + 5 + 5 + 5 = ?

You could just add them normally, treating each of those 5's as a size-5 step along the number line. But since each of these addition steps is the same size, a faster way to figure out the result would be to determine two things: the size of the step, and how many steps we have. Then we can multiply the size of step (in this case, 5) by the number of steps. In this case, we have a total of 6 size-5 steps, so we'd say:

5 * 6 = 30 Size of step (5) times number of steps (6) = total number (30).

Simple enough, right?

Well, exponents are a similar type of shortcut – except this time, it's actually a shortcut for the shortcut! Exponents are a shortcut for multiplying a bunch of the same number together, in the same way that multiplication is itself a shortcut for adding a bunch of the same number together. So, if we had:

2 * 2 * 2 * 2 * 2 = ?

Again, we could multiply them normally, but that'd take a while. Since all the steps are the same size, we can take a notational shortcut and write this as an exponent. The size of step is the base (2), the number of steps is the exponent (5). So this case could be rewritten as:


It's important to note that I said “notational shortcut” up there. This exponent has exactly the same value as that series of * 2's above it, which in turn has the same value as:

{[(2 + 2) + (2 + 2)] + [(2 + 2) + (2 + 2)] } + {[(2 + 2) + (2 + 2)] + [(2 + 2) + (2 + 2)] }

But that is nearly impossible to read, let alone to work with – it took me several minutes just to work out where all the brackets went! So we use notational shortcuts to express these concepts – which at their heart are just complicated forms of counting – in a way that's easier to work with. But it's important to note that even the most complex high-school algebra is just a fancy way of writing some complicated counting. You could take the long way around if you really wanted to, but it'd be much too confusing for everyday manipulations, so we come up with shortcuts.

But back to exponents. Knowing our shortcut makes it relatively simple to figure out the total number when faced with a problem like:

25 = x

You just take 2 * 2 * 2 * 2 * 2, and you arrive at 32 as your answer. Again, the base is the size of step, the exponent is the number of steps. The only difference is that these are what Vi Hart calls “times-ish” steps – steps where you're increasing the value not by adding, but by multiplying.

One of the basic truths of mathematics is that for every operation, there is a way to undo that operation. Want to undo a multiplication? Just divide. Want to undo an addition? Subtraction's got you covered. (This, incidentally, is sometimes why students find themselves going around and around in an infinite loop within a problem – at some point they're undoing an operation they've done elsewhere in the problem, and getting back to where they started.) To undo an exponent, generally taking a root will have you covered. For example:

We've arrived at 32 after 5 “times-ish” steps, but I want to know what size step we took to get there. Or, in math terms:

x5 = 32

Roots have got you covered here. Just find the 5th root of 32 and you'll have your answer:

5√32 = 2

But wait! There are actually three different positions in the basic exponent equation, right?

bp = r                                 b = base, p = power, r = result

We've seen how to solve if our variable is the result or the base, but what if the variable IS the exponent? In other words, what if I know the size of step and the number I want to get to, but not how many steps it would take to get there?

2x = 32

How do we undo THAT one?

This is a case where, because there are three parts to the original operation, we need a second way to undo it. THAT is where logarithms come in. Logarithms are the way to undo exponentiation to solve for the exponent itself.

bp = r                      solve for p?

Log b r = p

Traditionally, we read the equation above as “The log in base b of r is p.” When you hear the word “log,” think “number of steps to get there.” So you could really read the equation as “The number of steps of size b to get to r is p.”

Most math teachers will give you a funky-looking image with arrows pointing in a circle around the exponent to show you which variable to place where, however those images have always struck me as more confusing than they are helpful. The way I remember which number goes where actually clues off of a later part of the sentence: the phrase “in base b.” Just remember, the base is the original number, the number that says how big the steps are. The logarithm is your way of figuring out the number of steps to a total, so the total should be on the same side of the equation as your base. That leaves the exponent by itself on the other side of the equation, where we can solve for it easily.

So for our example problem from earlier:

2x= 32

log 2 32 = x

So now we know what we're doing. Generally we complete logarithms either by entering them on a scientific calculator or referencing a table, since it is very difficult to calculate the value of a logarithm by hand without using trial and error. Unfortunately this can lead to math teachers focusing on teaching students to simply memorize which numbers go where in the calculator, rather than actually teaching the theory behind what a logarithm IS and why we set it up the way we do. But hopefully you understand a bit more now than you did five minutes ago. Just remember, logarithms are nothing more than your way of undoing an exponent when the exponent itself is unknown.
You can check out Vi Hart's Logarithms video here.
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