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PROPERTY If m is positive integer then Elmr) (E(z))" _ In particular Elm) (E(L))'CALCULUS IN THE [7TH AND [STH CENTURIESProblem 17 Prove Property 3.PROPER...

Question

PROPERTY If m is positive integer then Elmr) (E(z))" _ In particular Elm) (E(L))'CALCULUS IN THE [7TH AND [STH CENTURIESProblem 17 Prove Property 3.PROPERTYE(-I) = ElT)(E(z))-

PROPERTY If m is positive integer then Elmr) (E(z))" _ In particular Elm) (E(L))' CALCULUS IN THE [7TH AND [STH CENTURIES Problem 17 Prove Property 3. PROPERTY E(-I) = ElT) (E(z))-



Answers

Prove Property 3 of Theorem 11

And that's probably want to give justification for the steps of the proof of part one of three. The first thing they tell us is to let X be the log base A of them. And then we're going to let why do the log base save in. Mhm. Now the first thing that we need to justify here is that M is equal to a to the X And N is equal to eight of the wine. And this just comes from the definition of the large. You were given those lawns there up top and we just invoke those definitions in order to get those statements. Now, next we have that em in is april to A to the X plus Y. And that's just the property of exponents whenever you multiply exponents that have the same base and both of these are both aids you add response. Mhm. Now next we say that the log base A of em in is equal to X plus why? And so we did here was tight the log base A. Of each side. Yeah. And then last week we can say that the log base A. Of them in is equal the world based Avian plus the log base A. Of in. And this is just by the transitive property of equality. We knew what X. Was equal to above. It was equal to the log base A. Of them. We knew why was equal to above. And that was a long based a event. And since both of those are equal to log by, save him in, they must be able to each other.

This problem. We are given the proof of part three of 73 And we would like to justify each of the given steps as we begin with X. Being able to the log base A. Of them and why being able to the log base A. Of the end. And so this is what we have been given. Now first they tell us that M. Is equal to A. To the X. And this is the first step in their proof. Well that's just invoking the definition of the law algorithm. We know that actually for the log base E. Of M. And that means that aided the X. Power as you pull it. Yeah. Now the next step that we are told is that the end of the K. Is equal to aid the act. It's okay. And that just took each side to the K. Power. What did he tied to the cave? Howard does maintain equality and that's what we did here on three we have thus the lord base A. Of them to the K. Is equal up to X cane. Okay. Which is equal to K. Times the Lord base A. Of em. And here we just took the log of each side from two, which maintains equality. And so we have reached the desired results. We've proven what we wanted to prove and we've justified what we need to to justify and so are proof has done.

Okay, So in this question, we want to show that the distributive properties off multiplication addition hold for said M. So, basically, we want to show this first line here, and these are something he's a couple of formulas that will have to remember. So are they. Will this former A a plus B is equal to this on a level as one and then truth would be a plus. M b is eager to aim what m plus and be more m and saving for some the dot So three will be this one and four will be this one. So first off, well, we will start off with a A, but and B plus by sea month. This is left side. So if we apply one to this on the addition, you have a dots Mm b plus by sea mud him. We'll be back around that. Then we will use four on this multiplication. Do you have a dot gods in B plus C now using three on this multiplication to expand it out or yeah, you have a tons by B plus by sea, and then you take mud and and then using regular distributive properties you have a B plus by a C mud and okay, that's no. We want to go from the left hand side. So a thoughts and be and susp I and a lot and see So then, using three ticks ban the dots. These bots here you have a being more than a be more than And that's why a C more than then using number two to expand this to fund. Plus, you have a baby plus m hey c then using one to expand this. Plus you have a B plus fire a C mother then So then this is your left hand side that's inside and this is your right hand side. So since your left hand side is equal, left hand side is equal to your right and sign it follows that this line right here must be equal to this. So therefore the distributive property holds over set

This probably would like to prove part two of them, three. It starts out by letting X. Equal to log base A. Of them and why equaling the log base A. Of and and that's just what we're given. But we want to justify each of the steps that they give us in the proof and the first is that eight of the X. Is equal to. Um And then eight of the Y. Is equal to in this just comes from the definition of the law and we take the logs that we were given up top and vote their definitions and that gives us that step there. Now in part two they tell us that M over in is equal to eight of the X. over eight of the Y. Okay. Which is equal to A. To the X minus Y. This is just the division property of excellence. Okay, we divided Eminem. We'll make that a capital in there. They were divided Eminem. And since M. Is A. The accent in the state of the why when we divide those, that becomes A to the x minus Y. We subtract the exponents because we're dividing them with the same base. Now from here we can say yeah, yeah with the Lord base A of em over in is equal to act minus why we just took the lot of each side and that maintains equality. And then lastly we can say that this is in fact equal to the log base A of em minus the Lord based A of in and this is from the definition of X. And why we said that X was equal to log base A of em. We said that why was equal to log base gave in and so we know the next minus Y is equal to the long, basically of them minus the law and they say it in and that is what we wanted to prove. Okay.


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