Many students (and parents) cower in fear when thinking about logarithms. However, while there are additional rules for dealing with these mathematical entities, there is nothing fundamentally different about exponentials and logarithms than there is about squaring and square roots. The problem is that most books don’t teach them well.
Let’s start by looking at an exponential formula:
a = bc
We will begin by naming each piece in this equation. We will call b the “base,” we will call c the “exponent,” and we will call a the “value.”
Now, let’s consider something we already know how to do—squaring. Here is a simple problem with squaring:
25 = x2
So, here, x is the base, 2 is the exponent, and 25 is the value. While you may be able to do this in your head, slow down and think about what is the proper operation to do to solve this problem? If you said, “square root,” you are correct. But why? Well, square root is the inverse operation (or what I like to call the “undo operation”) of squaring. If I square root both sides of the equation, the left-hand size becomes 5 and the right-hand side becomes just x.
Now consider the following:
64 = x3
Again, let’s name the pieces. 64 is the value, x is the base, and 3 is the exponent. So what is the correct operation? It is now the cube root, since x is cubed. Cube rooting both sides yields x because cube rooting is the “undo” operation for cubing.
So, we see a pattern here—to “undo” the exponent, we take the equivalent root. If we have xn, then we simply take the nth root.
Now let us turn to exponentials and logarithms. Exponentials actually have the exact same form, but with our unknown in the exponent position. Here is an example:
64 = 4x
Here, 64 is the value, 4 is the base, and x is the exponent. However, we can’t take the nth root of this, because, since the exponent itself is a variable, so we don’t know which root to take!
This is where logarithms come in. With rooting, we say, “given a particular value and exponent, what is the base?” With logarithms, we are saying, “given a particular value and base, what is the exponent?”
And, just like we had separate roots for every exponent (square root for 2, cube root for 3, etc.), we have different logs for every base. We use Log4 for a base of 4, Log22 for a base of 22, etc.
So, in our equation above, since 4 is the base, we will want to apply Log4 to both sides to “undo” the exponentiation and get back the exponent. This gives us:
log4(64) = log4(4x)
Since log4 is the “undo” operation for 4x, this will just leave x behind. This gives the equation:
log4(64) = x
We can then figure out log4(64) by using our calculator (the answer is 3).
The way to think about it is this: if I have bc, then a “root” is the undo operation that gives back the base by itself, and “logarithm” is the undo operation that gives back the exponent by itself.
There are additional rules to exponents and logarithms, and, oftentimes, I think these are introduced too quickly. First, a student needs to have a good grasp on what exponents and logarithms are before moving forward into their more interesting behaviors.
Logarithms sound scary, but if you teach them as being similar to roots, but for the base instead of the exponent, then it creates a stepping stone in the student’s mind which helps them move forward. It helps especially to have each part of the formula named so students are more clear about the relationship between the components.
