(This
is what happens when I have no internet. Post brought to you by Dunkin' Donuts free wi-fi.)
I've never really liked the way exponentiation is defined.
You start out with whole numbers, and that makes things easy.
Multiplication is repeated addition. “a * b” is “a” added to itself “b” times. (Which is also “b” added to itself “a” times.)
Exponentiation is repeated multiplication. “ab” is “a” multiplied by itself “b” times.
You move onto the rational numbers, then the real numbers.
Multiplication stays fairly straightforward. In fact, in an axiomatic approach, multiplication is one of the things we take as so basic that it's a given.
Exponentiation becomes “to determine the value of x to the y evaluate the natural logarithm at x, multiply the result by y, and evaluate the exponential function at that result.”
If this isn't setting off your “What the fuck?” alarms, it probably means you've already taken this class.
I say again: The expression xy is supposed to be “x multiplied by itself y times” and therefore to find out what xy is you evaluate a function, which has nothing to do with multiplication, at x, then multiply by y, and finally plug the result into another function that has nothing to do with multiplication.
Now, there is a reason for this. Two actually.
Before I state the reasons, let me convert that into a more mathy and less wordy form.
xy = exp(y*ln(x))
Now, onto the two reasons why that's the definition of xy.
The first is that the math checks out and it works just fine.
Specifically, the two functions are inverses so “xy = exp(ln(xy))” is definitely true (provided both sides of the equation actually exist), and the natural logarithm has the property that “ln(xy)=y*ln(x)” which means that the equality from the definition holds.
More importantly, we can solve it.
How is that more important than it being true?
We can make up infinite things that are equal to xy, but most of them don't help us answer the question of what xy actually is. This does.
We know the exponential. We know the natural logarithm. We know how to multiply. Thus converting “xy” to “exp(y*ln(x))” has changed something we didn't know how to solve (raising a number to an arbitrary power) to three things we do know how to solve.
So, that's why we do it. First off it's true, second it's useful.
For what it's worth, this is usually written as “xy = ey*ln(x)”.
So . . . “e”.
e is a very special number. For the moment, that matters to us not a bit. What matters is that “exp(1) = e” and, because the functions are inverses, “ln(e) = 1”.
So if you set the x in the “xy” we've been using equal to e, you get:
That's what justifies converting “exp([ ])” to “e[ ]”. We can't, however, start with that. ex is undefined until we define it using the exponential function, so we need the exponential function first. Once we define it, though, it's equal to the exponential function and can be used to stand in for it.
So “xy = exp(y*ln(x))”, usually written as “xy = ey*ln(x)”, undeniably works and it happens to be useful. It lets us solve xy for arbitrary values of y. But do a little comparison.
As previously noted, multiplication, our “repeated addition”, is so simple and basic we take it as a given. Exponentiation, our “repeated multiplication”, is so complex and exotic we have to invent two functions just to solve basic equations. And not just any functions. Difficult functions.
Difficult enough that my word processor (I'm not composing this online since I have no internet right now, that said my html-fu likely wouldn't be up to the task either) can't actually show what ln(x) is equal to.
So I'm going to have to use words instead of symbols. You take the integral of one over t dt. My word processor can show that (admittedly badly, but it can do it): “ʃ1/t ∂t”. Then you evaluate that integral from one to x. The result is ln(x).
Or, to put it another way, you have to invent fucking calculus before you can even begin to understand the definition of xy.
You have to invent calculus, integrate a function that resists simple integration, and when you've done that you're still only halfway there.
The exponential function doesn't exist yet. We can't just say it's ewhatever either, because that's not defined until we've created the exponential function.
So first you invent calculus. Then integrate one over t dt from one to x and call the result ln(x). Then you determine the inverse of the function ln(x) and call it exp(x).
Then, finally, you look at the number whose value you're trying to determine, xy. You evaluate ln(x) at that particular x, multiply it by that particular y, and take the exponential of the whole damn thing.
And only then do you know what xy means.
And that is why I don't like the way exponentiation is defined.
I've never really liked the way exponentiation is defined.
You start out with whole numbers, and that makes things easy.
Multiplication is repeated addition. “a * b” is “a” added to itself “b” times. (Which is also “b” added to itself “a” times.)
Exponentiation is repeated multiplication. “ab” is “a” multiplied by itself “b” times.
You move onto the rational numbers, then the real numbers.
Multiplication stays fairly straightforward. In fact, in an axiomatic approach, multiplication is one of the things we take as so basic that it's a given.
Exponentiation becomes “to determine the value of x to the y evaluate the natural logarithm at x, multiply the result by y, and evaluate the exponential function at that result.”
If this isn't setting off your “What the fuck?” alarms, it probably means you've already taken this class.
I say again: The expression xy is supposed to be “x multiplied by itself y times” and therefore to find out what xy is you evaluate a function, which has nothing to do with multiplication, at x, then multiply by y, and finally plug the result into another function that has nothing to do with multiplication.
Now, there is a reason for this. Two actually.
⁂
Before I state the reasons, let me convert that into a more mathy and less wordy form.
xy = exp(y*ln(x))
Now, onto the two reasons why that's the definition of xy.
The first is that the math checks out and it works just fine.
Specifically, the two functions are inverses so “xy = exp(ln(xy))” is definitely true (provided both sides of the equation actually exist), and the natural logarithm has the property that “ln(xy)=y*ln(x)” which means that the equality from the definition holds.
More importantly, we can solve it.
How is that more important than it being true?
We can make up infinite things that are equal to xy, but most of them don't help us answer the question of what xy actually is. This does.
We know the exponential. We know the natural logarithm. We know how to multiply. Thus converting “xy” to “exp(y*ln(x))” has changed something we didn't know how to solve (raising a number to an arbitrary power) to three things we do know how to solve.
So, that's why we do it. First off it's true, second it's useful.
For what it's worth, this is usually written as “xy = ey*ln(x)”.
So . . . “e”.
e is a very special number. For the moment, that matters to us not a bit. What matters is that “exp(1) = e” and, because the functions are inverses, “ln(e) = 1”.
So if you set the x in the “xy” we've been using equal to e, you get:
ey
= exp(y*ln(e)) = exp(y*1) = exp(y)
That's what justifies converting “exp([ ])” to “e[ ]”. We can't, however, start with that. ex is undefined until we define it using the exponential function, so we need the exponential function first. Once we define it, though, it's equal to the exponential function and can be used to stand in for it.
⁂
So “xy = exp(y*ln(x))”, usually written as “xy = ey*ln(x)”, undeniably works and it happens to be useful. It lets us solve xy for arbitrary values of y. But do a little comparison.
As previously noted, multiplication, our “repeated addition”, is so simple and basic we take it as a given. Exponentiation, our “repeated multiplication”, is so complex and exotic we have to invent two functions just to solve basic equations. And not just any functions. Difficult functions.
Difficult enough that my word processor (I'm not composing this online since I have no internet right now, that said my html-fu likely wouldn't be up to the task either) can't actually show what ln(x) is equal to.
So I'm going to have to use words instead of symbols. You take the integral of one over t dt. My word processor can show that (admittedly badly, but it can do it): “ʃ1/t ∂t”. Then you evaluate that integral from one to x. The result is ln(x).
Or, to put it another way, you have to invent fucking calculus before you can even begin to understand the definition of xy.
You have to invent calculus, integrate a function that resists simple integration, and when you've done that you're still only halfway there.
The exponential function doesn't exist yet. We can't just say it's ewhatever either, because that's not defined until we've created the exponential function.
So first you invent calculus. Then integrate one over t dt from one to x and call the result ln(x). Then you determine the inverse of the function ln(x) and call it exp(x).
Then, finally, you look at the number whose value you're trying to determine, xy. You evaluate ln(x) at that particular x, multiply it by that particular y, and take the exponential of the whole damn thing.
And only then do you know what xy means.
‧ ‧ ‧
And that is why I don't like the way exponentiation is defined.
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