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The formula $t=\frac{\log K}{0.045 \log e}$ gives the time $t$ (in years) that it will take an investment $P$ that is compounded continuously at a rate of 4.5$\%$ to increase to an amount $K$ times the original principal.a. Use the formula to complete the table to three decimal places.$$\begin{array}{|c|c|c|c|c|c|c|c|}\hline K & {1} & {2} & {3} & {4} & {5} & {10} & {20} & {30} \\ \hline t & {} & {} & {} & {} & {} & {} \\ \hline\end{array}$$b. Use the table to graph the function $t=\frac{\log K}{0.045 \log e}$c. If Paul invests $\$ 1,000$ in a savings account that is compounded continuously at a rate of $4.5 \%,$ when will his investment double? triple?

a. See tableb. See graphc. Double: about 15 years, Triple: about 24 years

Algebra

Chapter 8

LOGARITHMIC FUNCTIONS

Section 4

Common Logarithms

Functions

Exponential and Logarithmic Functions

Campbell University

University of Michigan - Ann Arbor

Idaho State University

Lectures

01:32

In mathematics, the absolu…

01:11

02:31

A principal $P,$ invested …

03:17

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Solve.Use $P(t)=P_{0} …

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The amount $A$ accumulated…

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04:21

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02:08

Suppose that $P$ dollars i…

02:48

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A sum of $\$ 5000$ is inve…

04:19

Hello, guys. So we have this problem here. It's a principal problem. Interest problem. Make more sense. Um, so in this interest problem, we have to figure out, um, we have to fill out this chart right here, using this time equation that were given. But what is the time equation for two? Well, um, like, what is the time? I mean, this case So according to the problem at the time, refers to how long it will take for your initial investment to multiply K amount of times. So it's hypothetical would be safe to say you invest $1000 right? And you want to find out. How long will it take for this to turn into $2000? Twice as much? Well, you would plug in to Miss Kay and then plugging the so formula. And then you should get this amount of time a size what it refers to so essentially, to fill out this graph, you just have to plug it into a calculator that many times. But this can be a little bit difficult toe plug into your caterpillar, depending what you have, So you might want to rewrite it. So I'm just gonna do this for the people who who need this to be rewritten to be a bit easier, um, Tiu to simplify it, you could, um, simplify the denominator. So you see how it has a coefficient here? It has a coefficient you can simplify by bringing it as an expert because one of the lager than little states that lug eight of parties say, Miss Love, um, is be times log. Hey, in this case, we could write it as a log e the power of 0.45 All right, And this here is a constant right. So you can just plug this into the calculator, having us the denominator and then plug in different case, and that should be easier. So I'm gonna plug in some values, and hopefully you should get the same one. So I cook in, huh? Log log. Can I put E to the power of 0.45? And for the expression, I get zero point 01 nine five or I'm going as far as I can. Um, in case you don't have an e, but in which I'm sure most of you do. But just in case you don't. According to my calculator, E is 2.7 18 two eight Casino. We have this constant, and we have the top lock, Kate. So we just have to plug in law, okay? Every time. So I'm gonna do that log and then the first case, one the bottom we put, um, her number, which is Sierra 0.19 fight for. And we get zero. Which makes sense, because this is saying that how long will it take for your, um, investment to be the same will take no time at all. So this checks out now. Go back to it and plug in the number two. It's F one. Um, Now it says three decimal places. I'm gonna write a three decimal places. It's gonna be Yeah, teen 0.4 zero. Bye. Eight. Since the fourth place is an age, that means I can round up the five, so I'll make it a six. I'll just take that. Get put it in there. Okay? So that should be your second. So this, um, units are in years, so that takes 15 years for this to duck, which is, uh, I guess pretty good's a penny. Watch your interesting but, uh, let's go for three. So go back looking three. And this time we get 24 point for 17 66 is greater than five second round up to seven and make it an eight. So that's, er next value, and it should be increasing because it doesn't make sense that you are well, investment would start decreasing for any reason because it's not outlined and issue this is it compounded. Continue sleep. So that implies it's always increasing. They should never decrease. Good way to double check. Go back plug in four. Now I get 30 0.8 one 166 is greater than five. You can place the one with two. So that's and four boot before Yeah, for five. Same thing. Dirty five 0.77 one and two. But two is not greater than five. We just don't need it, and it's for that now. I hope you see somewhat of a pattern here. It seems to be increasing less and less every time. That might be something relevant in real life problems, so it's always good to look at what the trenches. But now all of a sudden, we're jumping to 10. So make sure you don't accidentally do six. I would probably do that. Accident Me. So now we have 51.177 00 was obviously less than all fire. So you can get just get it. And that's definitely bigger number. But also realize that it's a bit big jump for K for to be 10 times and take 51 years. Um, just two more times six. Did you 6.5, Kate two 99 is definitely bigger than five. So that comes with three. Did you get that? You're different in last one. They're ready. And it is 70 five point bye. No, Uh, 47 seven is greater than five. Simon's weekend. We write that for as of, uh, there we go, who find we got it done. And that's party. So hopefully you see some kind of trend kind, you know, sort of what? The problems asking. And, uh, that's party. Now, a new party says that we have to make a table using the the points we have here, So, uh, yeah, I'm going to moves to this page right here. So we have to think about the values we have when we're graphing. So if we go back here, um, we have around 15. 24 30 35 51 66 75. Right. These air pretty wide range with numbers. And for K, we have, um, rather smaller set. So, um, you have to think what is the deep end? It's an independent variable. So this time depend on cater to skate depend on time. Well, obviously, time depends on Kate. Depending what the K is your time will be different. Getting what the time is. Your cave may be different, but not in this formula. So let's establish that at the beginning. This okay, this look his time. All right. So who so, um, for kay, we go 12 30. I'm depending on your grid. Um, you might two different intervals between them, but I'm going to to it in to soil. Let's let's see what we can do. 2468 10. 12 e. I don't think I can do to. So it's 25 Then if I might be too much for let's do for or should I zero for eight? Well, 16 one tea 24. 28. Yeah. Yeah, I think we're going to have to two and maybe extended a bit, too. For six, 8 10 Well, 14 16 18 20 two, 24 26 28 30. Okay, so we have our cake. We have all the way from 0 to 30 times. That's good. But the more complicated one it's gonna be t We have all these complicated numbers and it's gonna be kind of hard to fit them all in. But we have Ciro to 75 so I would think 0 75 we might want to do in terms of five. These are fairly close toe factors of five. So let's let's see. What if aphids go five, 10 15 Corny. Five, 30 35. 40 Already Fine. D five, 60 65 70. And what did we need to go up to? 75. So let's just add that up there. Okay, so that that mostly works hopefully have a need of one. But this will do. Which is the CIA General trends that are going on. So our first plot is fairly easy. It's 10 So you go one around here, zero up. So that's around there. Go yet one in the K zero in the time and you go back. You go to and around 15. Since we're in terms of five, getting pretty much on 15 is is good enough. So too, And 15 2 15 right around there. Then we have three and 24. That's just below 25. It's a three and 24 run there. Then four this 30 close to 31 so above 30 right above it, though then five is 35 more on 36. So above 35 just a little bit. And then we have 10 and 51. So a little. Oh, both decent tens over here fifties right there. Just a little bit above again. You don't have to be exact 20 and 66 maybe 67. So I'm gonna do it. A good chunk over 65. And that is for those who began 20. So 20 years, all the way over here, A good chunk of her 65 right around there. Let's see then last one is 30 with 75 around 30 with 75 way off there. So I hope you notice but it's kind of Ah, exponential curve or more of a log, a rhythm curve because its opposite of exponential curve. But it's it was first really fast and starts to slow down. If you see that curves like this, if I could drive, Yeah, ignore the curves. But but But I hope you see that trend, um, CIA. So that's part B. Um, yes. So it doesn't necessarily say to to scrap a line, just a graph, the plots. So I think you're supposed to kind of see the on the pattern now the final notice along problem. But we're at the final one. It says If Paul infests $1000 it's principal is $1000. When will he have his investment double and then also has tripled? So it's actually it's actually two questions. When does the Doublemint is the triple? So double would mean that cake was too right. Because, you know, two times triple means chemicals. Three. So we're just plugging in when Kate goes to and K equals three. So actually, the principal doesn't matter. You know why? Because it's not even in the equation. So this actually doesn't matter what it is It's just kind of giving you a real life example off. What, you know, that this could be applied for. So let's let's plug in the equation so poor back here. So we do, um, the log. But instead of, um, 30 or one this time we're going to put just to away Sorry, my bad. They already have it for you. So this wasn't always the time. They would have the graph right here. We have it so that when you double already tells you how long it will take and then what was the 2nd 1? It was asking triple. We already have it right here. So we already wrote the answers down, so that was rather sneaky of them, making us learn without us knowing. But I appreciate that. Um, yes. So that's that's the answer. Um, yeah, these these problems are very long seen Garrity's. But at the end of the day, these air these their skills you ought to know. And this, especially this longer than curve is it's very important when it comes to thio interest and banking stuff like that. So if you're at all interested in these types of curves, you need to know your longer themselves. What works. So I know that was a long one. But I hope you learn from that. And I hope you can do these type of problems in the future.

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