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Proof of the logarithm quotient and power rules

Sal proves the logarithm quotient rule, log(a) - log(b) = log(a/b), and the power rule, k⋅log(a) = log(aᵏ). Created by Sal Khan.
Video transcript
Let's see if we can stumble our way to another logarithm property. So let's say that the log base x of A is equal to B. That's the same thing as saying that x to the B is equal to A. Fair enough. So what I want to do is experiment. What happens if I multiply this expression by another variable? Let's call it C. So I'm going to multiply both sides of this equation times C. And I'll just switch colors just to keep things interesting. That's not an x that's a C. I should probably just do a dot instead. Times C. So I'm going to multiply both sides of this equation times C. So I get C times log base x of A is equal to-- multiply both sides of the equation-- is equal to B times C. Fair enough. I think you realize I have not done anything profound just yet. But let's go back. We said that this is the same thing as this. So let's experiment with something. Let's raise this side to the power of C. So I'm going to raise this side to the power of C. That's a kind of caret. And when you type exponents that's what you would use, a caret. So I'm going to raise it to the power of C. So then, this side is x to the B to the C power, is equal to A to the C. All I did is I raised both sides of this equation to the Cth power. And what do we know about when you raise something to an exponent and you raise that whole thing to another exponent, what happens to the exponents? Well, that's just exponent rule and you just multiply those two exponents. This just implies that x to the BC is equal to A to the C. What can we do now? Well, I don't know. Let's take the logarithm of both sides. Or let's just write this-- let's not take the logarithm of both sides. Let's write this as a logarithm expression. We know that x to the BC is equal to A to the C. Well, that's the exact same thing as saying that the logarithm base x of A to the C is equal to BC. Correct? Because all I did is I rewrote this as a logarithm expression. And I think now you realized that something interesting has happened. That BC, well, of course, it's the same thing as this BC. So this expression must be equal to this expression. And I think we have another logarithm property. That if I have some kind of a coefficient in front of the logarithm where I'm multiplying the logarithm, so if I have C log-- Clog base x of A, but that's C times the logarithm base x of A. That equals the log base x of A to the C. So you could take this coefficient and instead make it an exponent on the term inside the logarithm. That is another logarithm property. So let's review what we know so far about logarithms. We know that if I write-- let me say-- well, let me just with the letters I've been using. C times logarithm base x of A is equal to logarithm base x of A to the C. We know that. And we know-- we just learned that logarithm base x of A plus logarithm base x of B is equal to the logarithm base x of A times B. Now let me ask you a question. What happens if instead of a positive sign here we put a negative sign? Well, you could probably figure it out yourself but we could do that same exact proof that we did in the beginning. But in this time we will set it up with a negative. Let's just say that log base x of A is equal to l. Let's say that log base x of B is equal to m. Let's say that log base x of A divided by B is equal to n. How can we write all of these expressions as exponents? Well, this just says that x to the l is equal to A. Let me switch colors. That keeps it interesting. This is just saying that x to the m is equal to B. And this is just saying that x to the n is equal to A/B. So what can we do here? Well what's another way of writing A/B? Well, that's just the same thing as writing x to the l because that's A, over x to the m. That's B. And this we know from our exponent rules-- this could also be written as x to the l, x to the negative m. Or that also equals x to the l minus m. So what do we know? We know that x to the n is equal to x to the l minus m. Those equal each other. I just made a big equal chain here. So we know that n is equal to l minus m. Well, what does that do for us? Well, what's another way of writing n? I'm going to do it up here because I think we have stumbled upon another logarithm rule. What's another way of writing n? Well, I did it right here. This is another way of writing n. So logarithm base x of A/B-- this is an x over here-- is equal to l. l is this right here. Log base x of A is equal to l. The log base x of A minus m. I wrote m right here. That's log base x of B. There you go. I probably didn't have to prove it. You could've probably tried it out with dividing it, but whatever. But you know are hopefully satisfied that we have this new logarithm property right there. Now I have one more logarithm property to show you, but I don't think I have time to show it in this video. So I will do it in the next video. I'll see you soon.