4 Rules of Logarithms To Follow To Solve (follow these rules)

all right to be able to solve a problem like this there's a couple properties of logarithms that you're going to need to know all right now I might not have gone in order and I was just going off the top of my head from working through this problem and I know that's not a property logarithm that's a property of exponents but for this problem it's going to be helpful so I'm going to show you where all of these are going to take place when doing a problem like this you can see there's quite a little properties that's why when your teacher says like hey make sure you memorize these or write these down to your notes like you're going to want to have these right and sometimes it is helpful as like a little tip when you're taking a test or you know you're doing homework like just write these down on a scratch piece of paper so you have them next to you I used to tell my students like you know I always have a sheet of paper with like your rules and your formulas even if your teacher doesn't let you use them on a test then you at least have them next to you and you're just used to always you know reviewing them and looking at what they read because these are all important and if you don't understand why they work then I would encourage you to just make sure that you understand um how they are like look it up or you know check out other videos that I have on this explaining these rules but you're going to want to know these so let's use these to get to this the first rule I want you to understand is to use the power rule of logarithms we don't want numbers or anything in front here of this of our Ln right what we want to do is we want to rewrite that as a power so just to remember that this ladies and gentlemen is a negative one but it was a different top but it was a different same color so we want to be able to rewrite this as a negative one times right so now what I can do is I can rewrite that up top so now I can have okay so now I have an exponential equation E raised to the Ln of x to the negative first power right you can take whatever numbers in front rewrite it as a power equal to 3. now this is important because now this is written here as an exponential equation okay now we want to get rid of something as an exponential equation we don't like this base having a power so if we want to get rid of the power what we're going to want to do is take the log of both sides now you're going to take the log based on the base of whatever the base is here so when you have 10 you're going to take log base 10. if it's 7 you take log base 7. when you have e as your base you're going to want to take the natural logarithm of both sides so just like you know when we're solving equations whatever you do one side you do on the other side make sure you take the log of both sides okay so I'm just squeezing that in here so let's write this down a little bit cleaner now okay so I'm adding some parentheses you know you can add them in for the arguments if you want to so now we can kind of go to this one this property right so this says Ln of e raised to this power right whatever this is is just equal to whatever that power is so when you have the Ln the natural log because I remember remember guys Ln has a base e right so you have Ln base e or any logarithm of the base equal to its argument raised to another Power that's just going to equal to whatever that power is so when I have Ln basically right it's going to be raised to Ln of x to the negative one that's just equal this side is now equal to Ln of x minus 1. okay so now we have a logarithm equal to another logarithm logarithm equal to no logical again guys I don't care if it's base 10 I don't care if it's base seven in this case it's base e right so again when you have two logarithms that are isolated there's nothing else going on there's no like seven or you know plus five like when you have a logarithm equal to another organ the arguments are going to be equal to each other so now I can rewrite this as x to the negative first power is equal to A3 now here is where this property comes into play okay because a lot of students will kind of get confused and like crap how do I get rid of the negative one on top do I take like the logarithm or base X or something like that's what you did over here right you said whenever you got to get something off the power log both sides and that's true right but in this case we don't want to take the logarithm of Base X that means it's going to have a base of a variable so what we need to understand here is this rule so I technically should have probably written this like down here sorry so this the rule of exponents says that if you have X raised to a negative power that's just equal to the reciprocal of that of 1 over x to the A so that's going to be 1 over x to the power and now if I want to go ahead and solve this I got to get X off the denominator so what I'm going to do is multiply by X on both sides and then that's going to divide out I'll have 1 is equal to 3x and then divide by 3. so therefore X is now going to equal to a positive 1 3. so hopefully this video was helpful hopefully describing these properties gave you some insight and some value for this video and if it did I know you're gonna enjoy the next video