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# Using the logarithm change of base rule

## Video transcript

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 rewrite 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 is that if you think about how you might be able to change the base of the logarithmic or the logarithms with logarithmic expressions you might be able to simplify this a good bit and 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 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 maybe we were two 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 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 people's calculators or there might be logarithmic tables for base 10 so here you're saying the the exponent that I have to raise a to to get to B is equal to the exponent 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 proved it in in another video but now let's see if we can apply it so going back to this yellow expression this once again it's 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 this is going to be equal to 1 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 is base 10 log of four over log of B now if I divide by some from fraction or some rational expression that's the same thing as multiplying by the reciprocal so this is going to be equal to 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 before I just multiplied it by one and so we can go in the other direction now using this little tool we establish at the beginning of the video this is the same thing as log base four of B log base 4 of B so yeah we have a pretty neat result that actually came out here we didn't prove it for any values well though we have a pretty general be 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 have to raise B to to get to four and then here I have what exponent to have to raise four to to get to B now it might seem a little bit magical until you actually put some 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 and 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 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 2 now let's try to tackle this one over here so I have log base C of 16 times log base 2 of C all right so this one once again it might be nice to rewrite these each of these as 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 could 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 could write this as log base 10 of C log base 10 of C over over log base 10 of 2 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 and 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 and I'm going to be left with log base 16 so I can log base 10 of 16 over over log base 10 of 2 and we know how to go the other direction here this is going to be is it going to be the logarithm log base 2 of 16 log base 2 of 16 and we're not done yet because all this is is what power do I need to raise to 2 to get to 16 off to raise 2 to the I have to raise 2 to the fourth power we did the blue color to raise 2 to the fourth power to get to 16 so that's this is kind of a cool thing because in the beginning I started with this variable C it look like we're going to deal it with a pretty abstract thing but you can actually evaluate this this kind of crazy looking expression right over here evaluates to the number 4 in fact I've had to run some type of a math scavenger hunt or something this could be a pretty good clue for evaluating 2 for you know walk this many steps in the for word or something it would be pretty cool