Using the logarithm change of base rule | Mathematics III | High School Math | Khan Academy

So we have two different logarithmic expressions here, one in yellow and one in this pinkish color. What I want you to do, like always, is pause the video and see if you can rewrite each of these logarithmic expressions in a simpler way.

I'll give you a hint in case you haven't started yet: The hint is that if you think about how you might be able to change the base of the logarithmic expressions, you might be able to simplify this a good bit. I'll give you even a further hint. When I'm talking about change of base, I'm saying that if I have the log base, and I'll color code it, log base a of b, this is going to be equal to log of b over log of a.

Now you might be saying, "Wait, wait! We wrote a logarithm here, but you didn't write what the base is." Well, this is going to be true regardless of which base you choose as long as you pick the same base. This could be base 9, in either case. Now, typically, people choose base 10, so 10 is the most typical one to use, and that's because most people's calculators or there might be logarithmic tables for base 10.

So here you're saying that the exponent I have to raise a to, to get to b, is equal to the exponent I have to raise 10 to, to get to b, divided by the exponent I have to raise 10 to, to get to a. This is a really, really useful thing to know if you are dealing with logarithms, and we prove it in another video.

But now let's see if we can apply it. Going back to this yellow expression, this, once again, is the same thing as 1 divided by this right over here. So let me write it that way. Actually, this is 1 divided by log base b of 4. Well, let's use what we just said over here to rewrite it. So this is going to be equal to 1 divided by, instead of writing it log base b of 4, we could write it as log of 4 over log of b.

Now, if I divide by some fraction or some rational expression, that's the same thing as multiplying by the reciprocal. So this is going to be equal to 1 times the reciprocal of this, log of b over log of 4, which of course is just going to be log of b over log of 4. I just multiplied it by 1.

So we can go in the other direction now. Using this little tool we established at the beginning of the video, this is the same thing as log base 4 of b. So we have a pretty neat result that actually came out here. We didn't prove it for any values, although we have a pretty general b here.

If I take the reciprocal of a logarithmic expression, I essentially have swapped the bases. This is log base b: What exponent do I have to raise b to, to get to 4? And then here I have: What exponent do I have to raise 4 to, to get to b?

Now, it might seem a little bit magical until you actually put some tangible numbers here. Then it starts to make sense, especially relative to fractional exponents. For example, we know that 4 to the third power is equal to 64. So if I had log base 4 of 64, that's going to be equal to 3. If I were to say log base 64 of 4, well now I'm going to have to raise that to the 1/3 power.

Notice they are the reciprocal of each other, so actually, not so magical after all, but it's nice to see how everything fits together. Now let's try to tackle this one over here. So I have log base c of 16 times log base 2 of c.

Alright, so this one, once again, it might be nice to rewrite each of these as a rational expression dealing with log base 10. So this first one, I could write this as log base 10 of 16. Remember, if I don't write the base, you can assume it's 10, over log base 10 of c.

And we're going to be multiplying this by, now this is going to be, we could write this as log base 10 of c over log base 10 of 2. Once again, I could have these little tens here if it makes you comfortable, I could do something like that, but I don't have to.

Now this is interesting because if I'm multiplying by log of c and dividing by log of c, both of them base 10, well, those are going to cancel out. I'm going to be left with log base 10 of 16 over log base 10 of 2. And we know how to go the other direction here.

This is going to be the logarithm log base 2 of 16. And we're not done yet because all this is; is what power do I need to raise 2 to, to get to 16? I'll have to raise 2 to the 4th power. The blue color, to raise 2 to the 4th power to get to 16.

So that's a kind of cool thing because in the beginning I started with this variable c. It looks like we're going to deal with a pretty abstract thing, but you can actually evaluate this. This kind of crazy looking expression right over here evaluates to the number four. In fact, if I had to run some type of a scavenger hunt or something, this could be a pretty good clue for evaluating 2 to 4. You know, walk this many steps in the forward or something. It would be pretty cool.