Using the logarithm change of base rule

Video transcript

- [Voiceover] So we have two different logarithmic expressions here, one in yellow and one in this pinkish color. And what I want you to do, like always, pause the video and see if you can re-write each of these logarithmic expressions in a simpler way. And 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, or the logarithms or the logarithmic expressions, you might be able to simplify this a good bit. And I'll give you an even 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, log base A of B, this is going to be equal to log of B, log of B over log of A, over log of A. Now you might be saying wait, 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 nine, base nine in either case. Now typically, people choose base 10. So 10 is the most typical one to use and that's because most peoples calculators or they might be logarithmic tables for base 10. So here you're saying the exponent that 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 we'll see if we can apply it. So now going back to this yellow expression, this once again, is the same thing as one divided by this right over here. So let me write it that way actually. This is one divided by log base B of four. Well let's use what we just said over here to re-write it. So this is going to be equal to, this is going to be equal to one, divided by, instead of writing it log base B of four, we could write it as log of four, and if I just, if I don't write the base there we can assume that it's base 10, log of four over log of B. Now if I divide by some fraction, or some rational expression, it's the same thing as multiplying by the reciprocal. So this is going to be equal one times the reciprocal of this. Log of B over log of four, which of course is just going to be log of B over log of four, I just multiplied it by one, and 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 four of B, log base four 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, 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 four? And then here I have what exponent do I have to raise four 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 four to the third power is equal to 64. So if I had log base four of 64, that's going to be equal to three. And if I were to say log base 64 of four, well now I'm going to have to raise that to the one third power. So 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, now let's try to tackle this one over here. So I've log base C of 16, times log base two of C, alright. So this one, once again it might be nice to re-write these, each of these, as a rational expression dealing with log base 10. So this first one, 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 over log base 10 of C, and we're going to be multiplying this by, now this is going to be, we can write this as, log base 10 of C, log base 10 of C over, over log base 10 of two. Log base 10 of two. Once again I could have these little 10's here if it makes you comfortable. I could do something like that but I don't have to. And now this is interesting, cuz 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 and I'm going to be left with log base 16, sorry log base 10 of 16 over, over log base 10 of two. And we know how to go the other direction here, this is going to be, this is gong to be the logarithm, log base two of 16. Log base two of 16, and we're not done yet because all this is is what power do I need to raise two to to get to 16? We'll have to raise two to the, I have to raise two to the fourth power. We did it in the blue color. To raise two to the fourth power to get to 16. So that's, this is kind of a cool thing, cuz in the beginning, I started with this variable C, it looked like we were going to have deal with a pretty abstract thing, but you can actually evaluate 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 math scavenger hunt or something, this could be a pretty good clue for evaluating to four. You know walk this many steps forward or something. It'd be pretty cool.