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CCSS.Math:

welcome to this presentation on logarithm properties now this is going to be a very hands-on presentation if you don't believe that one of these properties are true and you want them proved I've made three or four videos that actually prove these properties but what I'm going to do is I'm going to show you the properties and then show you how they can be used so gonna be a little more hands-on so let's just do a little bit of a review of just what a what a logarithm is so if I say that a oh that's that's not the right let's see I want to change there you go let's say I say that a let me start over a to the B is equal to C so if we a to the B power is equal to C so another way to write this exact same relationship but instead of writing the exponent is to write it as a logarithm so we could say that the logarithm the logarithm base a of C is equal to B so these are essentially saying the same thing they just have different kind of results in one you know a and B and you're kind of getting see that's what exponentiation does for you and the second one you know a and you know that when you raise it to some power you get C and then you figure out what B is so they're the exact same relationship just made in a different way now I will introduce you to some interesting logarithm properties and they actually just fall out of out of this relationship out of our out of this relationship and the regular exponent rules so the first is that the logarithm let me do a more cheerful color the logarithm let's say of any base so let's just call the base let's say B for base log of them base B of A plus logarithm base B of C and this only works if we have the same basis so that's that's important to remember that equals the logarithm of base B of a times C now what does this mean and how can we use it or let's just even try it out with with some well I don't know examples so this is saying that I'll switch to another color let's make move my mom I don't know I never know how to say that properly let's make that by my my example color so let's say logarithm of base 2 of I don't know of 8 plus logarithm base 2 of I don't know let's say 32 so in theory this should equal if we believe this property this should equal logarithm base 2 of what well we say 8 times 32 so 8 times 32 is 240 plus 16 256 256 let's see if that's true just trying out this number and this is really isn't a proof but it'll give you a little bit of an intuition I think for what's going on around here so log so this is we just use our property this little property that I presented to you and let's see if it's if it works out so log base 2 of 8 2 to what power is equal to 8 well 2 to the third power is equal to 8 right 2 to the third power is equal to 8 so this term right here that equals 3 right log base 2 of 8 is equal to 3 2 to what power is equal to 32 let's see 2 to the 4th power is 16 2 to the fifth power is 32 so this is right here is 2 to the this is 5 right and 2 to the what power is equal to 256 well let's see well if you're a computer science major you'll know that immediately that a byte can have 256 values in it so it's 2 to the 8th power but if you don't know that you could multiply it out yourself but this is 8 and I'm not doing it just because I knew that 3 plus 5 is equal to 8 I'm doing this dependently so this is equal to eight but it does turn out that three plus five is equal to eight this may seem like magic to you or it may seem obvious and for those of you who it might seem a little obvious you're probably thinking well 2 to the third times two to the fifth is equal to two to the three plus five right this is just an exponent rule I believe exponent I don't know I don't know the names of things and that equals two to eight two to the eighth and that's exactly what we did here right on this side we had 2 to the third times two to the fifth essentially and on this side you have them added to each other and what makes a logarithm interesting is and why it's a little confusing at first and you can watch the proofs if you really want to kind of a rigorous not even you've my proofs want rigorous but if you want kind of a better explanation of how this works but this should hopefully give you an intuition for why this property holds all right because when you multiply two numbers of the same base right to exponent exponential expressions in the same base you can add their exponents similarly when you have the log of two numbers multiplied by each other that's equivalent to the log of each of the numbers added to each other this is the same property if you don't believe me watch watch the the proof videos so let's do a let me show you another another log property that's pretty much the same one I almost view them and say so this is log base B of A - log base B of C is equal to log base B of I R I am running out of space a divided by C that says a divided by C and we can once again try it out with some with some numbers I used to a lot just because 2 is an easy number to to figure out the powers but let's use a different number let's say log log base 3 of I don't know log base 3 of well you know let's make it interesting let's log base 3 of 1/9 - log base 3 of 81 so this property this property tells us that this is the same thing as well I'm ending up with a big number log base 3 of 1/9 divided by 81 so that's the same thing as 1/9 times 1 over 81 I'm two large numbers for my for my example but we'll move forward so let's see 9 times 8 is 7 20 right 9 times right 9 times 8 is 7 26 1 over 729 so this is log base 3 over 1 over 729 so what what does 3 to what power is equal to 1/9 well 3 squared is equal to 9 right 3 squared is equal to 9 so 3 so we know that you know if 3 squared is equal to 9 then we know that 3 to the negative 2 is equal to 1/9 right the negative just inverts it so this is equal to negative 2 right and then minus 3 to what power is equal to 81 let's see 3 to the third power is 27 so 3 to the 4th power so we have minus 2 minus 4 is equal to well we could do it a couple of ways minus 2 minus 4 is equal to minus 6 and now we just have to confirm that 3 to the -6 power is equal to 1 over 729 so that's my question is 3 to the minus 6 power is that equal to 7 1 over 729 well that's the same thing as saying 3 to the 6th power is equal to 729 because that's all the negative exponent does is inverts it let's see we could multiply that out but that should be the case because well we could look here but let's see 3 to the 3 to the third power this would be 3 to the 3rd power times 3 to the 3rd power is equal to 27 times 27 that looks pretty close you can you can confirm it with a calculator if you don't believe me anyway that's all the time I have in this video in the next video I'll introduce you to the last two logarithm properties and if we have time maybe I'll I'll do examples with the leftover time I'll see you soon