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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.

## Want to join the conversation?

• what grade is logs for?
• Logarithms are part of the High School Algebra Core Curriculum Standards
• I think I just gained several brain cells at
• who can prove this to me?? how it can be answer of 1?
1/(log_a AB) + 1/(log_b AB) = 1
• That's easy (but changing b to x since there is a subscript x character):
1/logₐ(ax) + 1/logₓ(ax)
= [ log(a) / log(ax)] + [ log(x) / log(ax) ]
= [ log(a) + log (x) ] / log(ax)
= log (ax) / log (ax)
= 1
Provided that both a and x are positive. It is undefined if either a or x is ≤ 0
• What's the point of proofs?
• The point is to prove that this rules are not made up and that they are true. Just as you could put in the numbers into a formula you could prove it by using the properties o a function/operator to find out how the numbers move.
• Where is the next video Sal mentions at ?
• By knowing the previous two properties (product and power), you could prove the quotient property this way:
log(a) - log(b) = log(a) + (-1)log(b) = log(a) + log(b^-1) = log(a) + log(1/b) = log(a * 1/b) = log(a/b)
• how would you solve: 12^log12(4)
• since 12 is being brought to a log power with a base the same as it, they cancel out. so 12^log12(4)=4 Keep in mind this only works because there are the same number in those two specific places.

To see that it works first set it up like an equation.

12^log12(4) = x

Now turn it into another log. so if it were 12^y = x you sould make it log12(x)=y. so here 12^log12(4)=x becomes log12(x) = log12(4).

x has to be 4 because if log12(x) = log12(4) log base 12 only gets to the number log12(4) equals when the number inside is 4, no other number can get the sam result. so x must be 4
• There are more logarithm properties than this, they should be added to this section.
• Indeed there are way more rules, at least the ones I studied at school
• Can anyone prove that logaX=-logaX using log law 3?
• If logₐ(x)=-logₐ(x), then logₐ(x)=0. That has nothing to do with logarithms, zero is the only number that is its own negative.
• why did he raise it to c power intead of multiplying both sides by c
• by raising the x^b=a to the c power, he was able to get the equality, log_x(A^c)=BC. BC was also in the previous equation Sal made, which let him connect log_x(A^c)=C*log_x(A).
note: he did not have to do the same operation to both sides, because each side held a separate equality, not an equation.
log_x(A)=B = x^B=A

## 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.