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# Intro to logarithm properties (2 of 2)

CCSS.Math:

## Video transcript

welcome back well I'm going to show you the last two logarithm properties now so this one and and I always found this one to be in some ways the most obvious one but don't feel bad if it's not obvious maybe we'll take a little bit of introspection and I encourage you to really experiment with all these logarithm properties because that's the only way that you'll really learn them and the point of math isn't just to pass the next exam or to get an A on the next exam the point of math is to understand math and so you can actually apply it in life later on and not have to relearn everything every time so the next logarithm property is if I have a times the logarithm base B of c5 a times this whole thing that that equals that equals logarithm base B of C to the a power fascinating so let's see if this works out so let's say if I have I don't know three times logarithm base 2 of 8 so this property tells us that this is going to be the same thing as logarithm base 2 of 8 to the third power and that's the same thing well that's the same thing as well we could figure it out but so let's see what this is three times log base what's the log base 2 of 8 I the reason why I kind of hesitated the second it goes because every time I want to figure something out I want to use log and exponential rules kind of to do it so I'm trying to avoid that but anyway going back so so what is this log bit 2 to what power is 8 well 2 to the third power is 8 right so that's 3 and we have this 3 here so 3 times 3 so this thing right here should equal 9 if this equals 9 then we know that this property works at least this example you don't know if it works for all examples and for that maybe you'd want to look at at the proof we have in the other videos but that's kind of a you know more advanced topic but it you the important thing first is just to understand how to use it so let's see what is 2 to the 9th power well it's going to be some large so actually I know what it is it's a it's 256 because we just in the last video we figured out to the 8th was equal to 256 and so to the 9th it should be 512 so this so 2 to the 9th should be 512 so if 8 to the 3rd is also 512 then then we are correct right because log base 2 of 512 is going to be equal to 9 right well it's 8 to the third it's 64 time right 8 times 8 squared is 64 so 8 cubed so let's see 4 times 2 is 3 6 times a it looks like it's 512 correct and there are other ways you could have done it because you know you could have said 8 to the third is the same thing as 2 to the 9th how do we know that well 8 to the third is equal to 2 to the third to the third right I just rewrote 8 and we know from our exponent rules that 2 to the third to the third is the same thing as to the 9 and actually it's this exponent property where you can multiply when you take something to exponent then take that to an exponent and you can essentially just multiply the exponents that's the exponent property that actually leads to this logarithm property but I'm not going to dwell on that too much in this presentation there's a whole video on kind of proving it a little bit more formally the next logarithm property I'm going to show you and then I'll review everything and maybe do some examples this is probably the single most useful logarithm property if you are a calculator addict and I'll show you why so let's say I have log log log base B of a is equal to log base C of a divided by log base C of B now why is this a useful property if you are calculator addict well let's say you go to class and there's a quiz the teacher says week you can use your calculator and using your calculator I want you to figure out the log base 17 of 357 and you will scramble and look for the log base 17 button on your calculator and not find it because there is no log base 17 number a button on your calculator you'll probably either have a log button or you'll have an Ln button and just so you know the log button on your calculator is probably base 10 and your Ln number your Ln a button on your calculator is going to be base e for those of you who aren't familiar with you don't worry about it but it's you know 2.71 something something it's a number it's nothing you know that it's an amazing number but we'll talk more about that in a future presentation but so so there's only two bases you have on your calculator so if you want to figure out another base logarithm you use this property so if you if you're given this on an exam you can very confidently say oh well that is just the same thing as you would have to switch to your yellow color in order to act with confidence log base let's say we could do either E or 10 but you could say that's the same thing as log base 10 of 357 divided by log base 10 of 17 so you literally could just say you know type in 357 into your calculator and press the log button you're going to get by modal BAM then you know you can clear it or if you do not use parentheses and you can't get you can do that but you say you know you press type 70 in your calculator press the log button you go to member them and they just divide them and you get your answer so this is a super useful property for for calculator addicts and and once again I'm not going to go into a lot of depth of how this this one to me it's the most useful but it doesn't it's not it doesn't completely fall it does fall out of obviously of the exponent properties but it it's hard for me to describe the intuition simply so you probably want to watch the proof on it if you don't believe why this happens but anyway with all of those aside this is probably the one you're going to be using the most in everyday life I still use this in my in my job so you know the logarithms are useful let's do some let's do some examples so let's see let's just let's just rewrite a bunch of things in simpler forms so if I wanted to write I don't know the log base two of the square root of mmm let me think of something of 32 divided by the divided by the cube divided nihilistic the square root divided by the square root of eight how can I rewrite this so it's reasonably not messy well let's think about this this is the same thing this is equal to if I move vertically or horizontally but I'll move vertically this is the same thing as the log base two of 32 over the square root of eight to the one-half power right and we know from our logarithm properties the third one we learned that that is the same thing as one half times the logarithm logarithm of 32 divided by the square root of eight right I just took the exponent and made that the coefficient on the entire thing and we learned that in the beginning of this video and now we have a little quotient here right logarithm of 32 divided by logarithm of square root of eight well we can use our other logarithm let's keep the one half that's going to equal whoops parentheses logarithm I forgot my base logarithm base 2 of 32 - right because this isn't a quotient - the logarithm base 2 of the square root of 8 right let's see well here once again we have a square root here so we could say that this is equal to 1/2 times log base 2 of 32 - this is 8 to the one-half which is the same thing as 1/2 log base 2 of 8 we learned that property in the beginning presentation and then if we want we can distribute this original one-half and this equals 1/2 log base 2 of 32 minus 1/2 minus 1/4 it's going to distribute that 1/2 minus 1/4 log base 2 of 8 this is 5 halves - this is 3 3 times 1/4 minus 3/4 or 10 fourths minus 3/4 is equal to 7/4 I probably made some rhythmic arithmetic errors but you get the point see you soon