Calculus: Evaluating Logarithmic Functions

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Special Functions
Logarithmic Functions
Evaluating Logarithmic Functions Page [1 of 3]
It's now time for us to brave the frontier of logs. And in fact, I've prepared this lecture in advance, as you can see. We didn't want to waste time. Logs, there it is. Now, there's just one sentence that you need to know--just one sentence. Four words. And with those four words, you will never, ever get stuck with logs. And I'm going to write that sentence for you right now, here, and the sentence is: A log is an exponent.
I guess it's actually five words. Sorry about that. I wasn't counting the a. If you remember this simple sentence, your log problem days are over. Now, let me try to show you exactly what this sentence means. If I write log and write this symbol here, what does that cryptic mathematical symbol mean? Well, all you have to do is remember the chant, "a log is an exponent." And here's how you read that. A log is an exponent.
That means that c is the exponent that I have to raise this base d to in order to get a . Let me say that again. This says log base b of a equals c. Log base b of a equals c. What does it mean? Log is an exponent. So, this is the exponent I have to raise the base to in order to get a. So, this statement is identical--it says the exact same thing as b raised to the c power equals a. These two statements are identical, and let me show you why. Because a log is an exponent. And notice that c is the exponent I have to raise d to in order to get a.
If you just remember that little phrase, you will always be able to get yourself out of any log jam you get yourself into. Heh heh, log jam. I thought that was a goody. All right, let's take a look at some examples just to get our feet wet in this. In fact, why don't you try these? Log base two of eight. Log base two of one-fourth. Log base two of two. Log base two of one.
What I'd like for you to do is to try to make a guess as to what you think each of these equal--and remember that a log is an exponent--and see if you can figure out what these answers should be. Okay, give it a try right now.
How'd you make out? Let's take a look at them and think to ourselves what exactly is going on here. So, in the first question, what does that equal? Well, whatever that equals, I can think of it as a question mark. Log base two of eight equals a question mark. And I just remember the phrase, "Log is the exponent." So, question mark is the exponent that I have to raise two to in order to get eight. This is saying two to the question mark equals eight.
Well, so what's the question mark have to be. Well, if you think about it, I think the question mark has to be three, because two times two times two gives me eight. So, the answer to this question would be three, because two cubed equals eight.
What about this one? This is actually pretty hard, here. What would that one be? Well, if you think about that for a second and think to ourselves, "Well, this thing is the exponent I have to raise two to in order to get a fourth." Well, that sounds sort of weird. How can you raise two to a power, and get the number a fourth? Let's think about that. Two to the question mark equals a fourth.
Well, if I put in a one or a two or a three or a four, those numbers get bigger and bigger and bigger. They're getting further and further from a fourth. To make that number small, I have to sort of think about taking a reciprocal. Oh, then I remember that taking a reciprocal--I can do that in terms of powers by looking at a negative power. In fact, two to the negative one would be one over two. So, two to the negative two would be one over two squared, which would be a fourth.
So, this question mark must be negative if this number is a fourth, and two to the minus two equals one over two squared, which is, indeed, one-fourth. So, therefore, this answer must be negative two.
What about log base two of two? Well, what power do you have to raise two to get two? Lot of two's there. But the answer is one. Two to the first power is two, so this is just a one. And two to what power will give you one? The answer there is the zero power. In fact, there is a whole bunch of log questions that you can see these represent the exponents that I have raise those individual bases to in order to get these answers.
So the log untangles--if you will--exponentiation. In fact, let's try to graph this log function right now. If we graph this log function, what would we see? So, if I graph this function--well, I'm going to graph the function. I'll write it over here, so we can see it. F of x--right here--equals log base two of x. And we have a whole bunch of points we've plotted.
For example, if I plug in one, I see the value is zero. So, at one, I'm at zero. So, at one--I'm at zero. At two, I'm at one. So, when I go over two, I go up one. What does this say? At a fourth--if that's one, this would be a half, and so that would be a fourth--I see the value is actually negative two. One, two. And this value at eight, I go way over eight. I mean, eight is so far away, and look how high I climbed. Only three. I climbed not that high at all.
And, if I now connect them, I get this as the general graph of a log function. And you'll notice--you may notice--that in fact, this looks a little bit like the exponential function. In fact, if you turn your head a little bit, you will see the exponential function. I'm going to actually show you the exponential function there. There's the exponential function. See it?
So, the log and the exponential function turned out to have this inverse function relationship. One function is the inverse of the other. Let me say explicitly what that means. What that means is if I take log base b of b to the a power, what would that equal? Now, actually, this is sort of a mathematical tongue twister. You've got to sort of think about the little sentence, "a log is an exponent," and then figure out what this equals.
I'll say it for you, but I'm warning you right now, it's a tongue twister. It's not hard. It's a tongue twister. What is this equal to? Well, log is an exponent. So, this is the exponent I have to raise b to in order to get b to the a. So, what exponent do I raise b to get b to the a? The answer is "a." You see how it's a tongue twister? You've got to think to yourself, "B to what gives me b to the a?" Well, the answer is a. B to the a gives me b to the a. It's a mathematical tongue twister.
This identity is very important, and it's so confusing until you think to yourself long and hard how, in fact, a log is an exponent. This is the exponent I have to raise b to in order to get b to the a. Well, b to the a--a is the exponent, so this equals that.
Another useful fact, which is related to this, is if I take b and raise it to the log base b of a, what would that equal? Well, let's think about that. This is another confusing mathematical tongue twister. This is b raised to an exponent. What does it equal? Well, what is this exponent? Well, a log is an exponent. This is the exponent I have to raise b to in order to get a.
So, what is b raised to the exponent I have to raise b to in order to get a? The answer is a. Again, you've got to sort of replay this and think about what this is saying. B raised to the power that I have to raise b to in order to get a, gives me a. B raised to the power that I have to raise b to to get a gives me a. You can say it a lot--it's a mathematical tongue twister--but it turns out, it's also a mathematical identity. These are two very important relationships that link exponents to logarithms. And this shows the inverse--proportionately inverse--relationship between the log and the exponent.
Now, let me tell you a couple properties about logarithms that are important. Since a log is an exponent, all the properties we know about exponents go for logarithms. For example, if I have log base b of a product, then if we're multiplying, what do we do with the exponent? We add. So, log of a product is actually the sum of the log. And conversely, if I have log base b of a plus log base b of c, that equals log base b of the product of those things.
Again, because log is an exponent. Remember, back in the exponent section, we saw that n to the a times n to the b equals n to the a plus b. Well, that's to be sort of the a plus b thing here, and then we took the product. So, again, since it's an exponent, we add exponent.
Similarly, log base b of a divided by c--you probably guessed this--it's going to be the log base b of a minus--because when we take quotients, we subtract exponents. So, we subtract the log, because a log is an exponent.
So, these two facts are extremely useful and well worth remembering. In fact, more generally, you can generalize this kind of result by looking at another fact about exponents and then looking at how it relates to the log. If we have log base b of any number a raised to the c power. What happens if you take something, raise it to a power, and then take the whole thing and raise it to another power? What do you do with the exponents? You multiply them.
So, in fact, this exponent can be multiplied by the exponent of the log. In particular, this c can come out in front. This would equal c times log base b of a. That is to say, if I have log of a to a power, that power can be pulled out in front of the log. These two things are equal by properties of the exponent. This is another one that's going to be extremely important.
Now, I want to show you a whole bunch of formula, none of which are correct. These are all false. The first one is log base b of a plus b equals log base b of a times log base b of c. Oh, sorry, this should be a c. It's even more false. It's really false.
This formula, which people do a lot, is confused with the original formula. In fact, could someone hand me that original formula please? With the original formula, which says that the log of a product is the sum of the log--is the sum of the log. Oh, thank you. So, this is being confused with this formula right here. You have to remember that the formula is log of a product equals the sum of the log. This is a false formula.
This formula says the log of a sum is something. The log of a sum--there's nothing for this. There's nothing here. This is false. Another good one to see is the following. Log base b of a divided by log base b of c equals, and then some stuff--anything. A lot of things are here. There's no formula for this kind of thing. There's no formula for this kind of thing. We shouldn't be thinking about that. Again, that's being confused with the fact that the log of a quotient is the difference of logs. Notice how that is different than this. This is the quotient of logs. This is a log of a quotient.
You want to make sure that you get these false identities in your head to make sure you never make these mistakes. The only identities you want to use are the ones that I just showed you that are correct. So, this is an important thing to remember not to do these things.
Now, the last thing I wanted to tell is just a couple little words about notation. And the first notational thing I want to tell you is that if we don't write any base at all. So, if I just write--for example--this. See, I didn't write any base down there? I wrote no b? Well, that means that the base must be base ten. It's understood that it's base ten.
Now, what about if I want to look at the log of that very special number e--very special number e--that two point seven one eight two eight da da da da da da. That special number e that has the property that if you look at the graph of an exponential at x equals zero, that slope of the tangent line is equal to one. That special function e, such that e to the x has its own derivative, e to the x.
Now, then, if I want to write that special log, we have notation. Since it's so natural, since it's part of growth, it all tells how things change. It's so natural, we call this the natural log and write this as In a. So, ln a is just a shorthand way of writing log base e, where this is that special e--two point seven one eight two eight one--and it goes on.
So, some of the formulas that we've already seen, let me just recap for this. Natural log of a product is the sum of a natural log. I'm just reminding you of these formulas that I'm using this notation now. Remember, in each of these cases it just means log base e of these things, so it's the same formula, but I'm just trying to show them to you as we'll use them. Log of a divided by b equals log of a minus log of b. Log of a raised to the b power. I could bring that b out in front and write this as b log of a.
Natural log of e to the x, if I bring that out in front, I just see x. Because remember, that's the special number e, and so these two things cancel each other out. Think about it for a second. If I bring that out in front, I have the x there, and I've got to multiply it by natural log of e. So, what is the natural log of e. It's log base e of e. Log is an exponent. What's the exponent I have to raise e to in order to make it equal to e? The answer is one, so in fact, I just have x times one.
And then lastly, what I want to tell you is that e to the natural log of x again equals x, like you saw before. So, these are the formulas that we're gong to use a lot of, but I'm writing them out in terms of this very special base e, and now the new notation for the log of that. The log of that is a natural log.
Okay. Well, that now empowers us with all the notation--all the background--that we were going to need to talk about logarithms. Now, it's time to ask the question, "What's the derivative of the natural log?" And if you think about it for a second, remember that the derivative of e to the x is itself e to the x. But, what in the world could the derivative of an actual log be? We'll see.