Stony Brook MAT 123 Fall 2015
Lecture 17: Exponential growth/decay problems and logs
October 21, 2015

Start   okay so webassign problem we had the other day is on january 1st deposit 1 cent on january 2nd, deposit 3 cents january 3rd deposit 9 cents which you could also think of as 3 cents squared. january 4th youre gonna put .09
0:31tripled right?
dont think of that formula squared, excuse me thats... put in one cent the next thing you put in you triple it so you can think of that as .01, times 3 then you put in 9 cents you can think of it as .01 times 3, times 3 again or .01
1:00times 3 squared this is what makes it exponential next day .27 seconds instead we can say .01cents times 3 cubed each day were multiplying by 3, the number you put in the previous day. the states are exponential so 1 cent times 3, 1 cent times 3 times 3 1 cent times 3 times 3 times 3 so on the 10th day
1:32your gonna put in 1 cent 3 times 3 times 3, 9 times now you just have to know what 3 to the 9 is which you can find with a calculator if this were a test question this would be the answer from the 30th day thats a lot of money 3 to the 30 is a very big number okay, everyone understand how to do that on webassign?
2:02okay with that one another type of thing we had was was i doing half life problems last time?
you were doing half life in chemistry right?
half life no you didnt do that in high school chemistry? maybe do that in physics so you have radioactive elements they change, they decay
2:31so you have the nucleus and the nucleus is unstable and what happens is some of the elements in the nuclei break down into different nuclei and this is called radioactive decay you can omit alpha, bata, or gama partical it doesnt really matter you get through this lecture what happens is you can look at the portions you had at the beginning and what you had at a later amount of time you need to figure out how much time has pasted so thats called the half life
3:02half life is how long it takes to get half of what you had before so carbon date carbon 14 is radioactive and turns to the carbon 12 you know what the 14 and 12 stand for right?
so what happens is it changes and and it has a certain amount of time so the half life make up some element and lets say the half life
3:34is 5 days so what fraction or what percentage will be remaining after 20 days
4:03when i ask for percentage, what you should do is start with a 100 day 0 i have 100 whatever grams doesnt matter cause ill be at percent, 100 percent so after 5 days i have half as much so i have 50 or as i said before 50 right?
so you got to figure it out
4:31so after 10 days i have 25 lefte thats what the half life means, after 15 days i have 12.5 per cent so what we can do is measure percisly how much carbon you had before and how much you had after the carbon by the way is only good for
5:0050 thousand years or something 100 thousand years, its not really good till a million very multi million years old, then you use different elements always take a break down so half a uranium is half of the earth so you can use uranium more easily 20 days youre down to 6.25 so what percent remains at the 20 days 6.25 percent
5:31so another way to think of this is you can say the amount you have is 100, percent is 100 times 1/2 t/5, so why is it t/5?
when t is 5, you get a 1/2 of 1 when t is 10 you get a 1/2 would be 10/5, 10/5 is 2 so you get a half of a half
6:00that make sense?
when t is 15 15/5 is 3 so you get a 1/2 of a 1/2 of a 1/2, a 1/2 cubed it makes it an exponential formula just like one of those exponential formulas well if you were given a half life in years then t would be measured in years so you want to make sure you match whatever the units are
6:31want to match the units what ever the half life is measured in if its measured in nano seconds, or measured in years the t should be a multiple of a half so when t equals the half life you cut it in half once, since you double a half life you now have 1/2, 1/2 which is a 4th so general way to do half life
7:04i assume you did this in chemistry the amount that you have times the is the amount you start with times 1/2 t/k k is the half life
7:55we love pre calc?
no love no love for pre calc either
8:15so you can do a half life question, something like this can easily show up on the exam i could say im gonna cover this up
8:30les see
9:25the half life of an element is 6 hours, so theres 400 grams originally the amount you would have woulld be 400
9:32times 1/2 to the 10 over 6 do you know what 1/2 to the 10/6 is?
no you dont i mean how are you suppose to figure that out?
i dont know how youre suppose to figure that out i mean its 1/2 raised to the 5/3 so you take the cubed root of 1/3 and then you raise it to the 5 this is not fu, this is what calculators are for
10:02so if we gave a question that looked like that, thats all we expected just show you know how to find the half life of course you can take this and make it a slighly harder problem we all go that um you just have to write that so
10:37a new variation of this
11:38originally you have 180 grams of an element and after 6 hours, you have 40 grams of the element, how many would you have after 20 hours?
so this is not quite a half life, but you can do a half life but its easier to do it as an exponential problem so lets say how we can do it
12:03you dont have to do it for the 1/2 if you dont want to you can do it instead with a and b so a nice way to do exponentials, all exponentials are the form y equals some a times some b to the x, or t, whatever number you love a and b are constant, so we can use this information to figure out a and b
12:30so originally you have 180 grams of an element, and after 6 hours you have 40 grams how many do you have after 20 hours, well that tells you that after you originally have times 0 is 180 and then times 6, you have 40 so thats a pair of coordinates, so this is like the homework problem so now we can figure out what a and b are from the coordinates
13:02so when x is 0 y is 180, so 180 is a times b to the 0 but anything to the 0 is 1 so 180 is a so now you can rewrite the equation, so y is 1180 times b to the x okay does that make senes?
so agin, when be is 0 you have 180 plug in 0 and 180, anything to 0 is 1
13:30so now you now know a so go back to your equation and replace a with 180 we got half of the equation done now you say when x is 6 i have to get 40 so 40 is 180 times b to the 6th here comes the messy part divide by 180 you get 40/180 which reduces to 2/9
14:00b to the 6 so b is 2/9 to the 1/6 which you could find in your calculator, however you could just leave it to the 1/6 cause thats what we expect you to do we dont expect you to figure out what that is if you cant do it without a calculator so now your equation is 180 times 2/9, to the 1/6
14:33to the x or y is 180 times 2/9 to the x/6, can you guys see that over there?
ill rewrite that last part y is 180 times 2/9 to the 6th, that would be and this raised to the x
15:00so thats saying 180 times times 2/9, you can multiply these together to the x/6 now you just plug in 1 remember youre not expected to figure out what that number is we dont care 2/9 20/6, here we go get a calculator, you can find that exactly without a calculator i would find that exactly
15:30we have to give you nice even multiples we have to make this if i did 24 hours, it would be easy alright cause at 24 hours you would just have 24/6 is 4 is 4, so you would jsut do 180 times 2/9 to the 4th, its annoying but its doable but were not gonna do anything that messy, its not necessary lets do another one of these to make sure you understandit
16:02we go the idea some of you are nodding some are saying ugh seriously?
16:41originally you have 180 grams of an element and after 10 hours you have 60 grams hhow many would you have after 30 hours?
originally you have 180 grams of an element, after 10 hours its 60 grams, how many do you have after 30 hours?
so you say well, originally
17:00i have 180 grams so that means i originally have the point 0,180 and then i would have 10,60 so we figured out before so 180 a times b to the 0 b to the zero is one, because anyhting to the zero is 1 so a is 180'
17:33so my equation can be written as 180 times b to the x so far so good now that i know t equals, time equals 10 i get 60 so 60 is 180 times b to the 10 dont let the 10 scary you, thats what helps you figure out the fraction power
18:00divide by 180 and you get a 1/3 is b to the 10 so b is 1/3 to the 1/10 thats how you get the k in the denominator in the half life so that tells you that the y is 180 times 1/3 to the 10th
18:32to the x, y is 180 times 1/3 to the x over 10 couple of danger things dont multiply the 180 times the 1/3 the 1/3 has to get raised to the power PEMDAS dont d 180 times 1/3 thats 60, and thats not the problem only whats in the first part, we all say what the brawl?
now i say how many after 30 hours so i have y=180
19:01times 1/3 to the 30/10 whats 30/10 3 so it would be y is 180 times 1/3 cubed and you can find 1/3 cubed ah you know how to get the x over here?
so the x times the 10, thats over 10 now i wanna know how many i have after 30 hours
19:31so i plug in 30 for x see why im plugging in 30 for x?
cause i want to know how many i have after 30 hours so once i get to here i have my equation and then i can figure out how many i have after so many hours cause all i have to do
20:00is plug in for x and now it becomes a half so thats 100 hours i just plug in x i get 100 after 17 hours, i get 17/10 so the idea is by using the first 2 pieces of information i can get my general eqaution once i have that, i can get anything i want this is the final answer
20:31which you can then make 180/27 you can leave it like this, thats fine with me i dont think theres a difference, these are both correct you would really want to see, us to get this what we are really trying to figure out is once you figure out the formula anyone can plug in the numbers and graph, thats not the hardest part
21:00once you get to here its the easy part, the calculation is annoying but were not asking you to do the calculations were asking you to set it up okay lets have you practice one on your own
22:25try that one
22:35just like the last one first we know originally so times 0 we have 24 and at time 2, we have 20 so we use that information and just come up with a general equation
23:00so we have 24 to start 24 is a times b to the 0 very small 0 if youre watching on camera and anything to the 0 is 1 so 24 is a y is 24 times b to the x so we can just know if we are given the starting amount the original amount thats what a is a is the starting
23:31amount, you initial amount now after 2 hours im gonna have 20 20 is 24 times b squared b to the 2 divide by 24 and you get 5/6 you can certainly leave that as 20/24 equals b squared so 5/6 will be 1/2 b. or you couldve written the square root of 5/6
24:00theyre both fine that tells us that our equation is is y so y is 24 times 5/6 to the half
24:30times x or 24 times 5/6 to the x/2 because you multiply x times 1/2, so you get x/2 now you just need to find out how much is left at 7 hours s y=24 times 5/6 7/2 easy? were people able to do that one?
happy? yes?
25:09i know y=a times b to the x so i plug in 24 for a now i found out that b is 5/6 to the 1/2 so i take b and replace it to 5/6 to the 1/2 this gets plugged in like this since 1/2 to the x, i multiply the powers thats why i get x/2
25:31cause its a half of x you wouldx have this you could have that the radical is b over 2 question is if you have the radical 5/6, is that to the x/2 not thats to the x
26:02the radical means to the 1/2 power right and you write x to the 6 so this could be 24 square root of 5/6 to the 7 how d you feel about these?
loving these?
why 7? i wnat to know how many after 7 hours
26:50why am i doing 1/2 times x here and then doing x/2 i did that in the other one didnt i do that in previous ones?
i did right?
27:05you can hate me its fine
28:06originally you have 60 grams of an element and 3 hours later you have 50 grams of the element in other words how long does it take to get down to 30 grams this is not the same, its similar you still want to set up the equation
28:34but you might might have to use logarithms might not originally theres 60 grams and 3 hours later yo have 50 gram whats the half life?
29:00couple ways you can do this and no you dont have to use logs originally you have 60 and after 3 hours you have 50 we want to know what time will we have 30 thats what youre trying to figure out cause 30 is half of 60 so we know y is a times b to the x now we know
29:30that when 0 is gonna come out 60, right everybody feel comfortable with that?
thats one more time 60 is a times b to the 0 60 is a why do i cross out b to the 0?
cause anything to the 0 is 1 so my first step y=60 times b to the x then i say alright well i get 50 after 3 hours
30:00so 50 is 60 b to the 3, cubed so i divide and you get 5/6 b cubed so you can take cubed root, or you can raise it to the 1/3 power so 5 6 to the 1/3 b now my equation is y 60
30:30times 5/6 to the 1/3 to the x or 60 times 5/6 to the x/3 so far so good?
how did we do on that part?
now we just have to plug in, now we have to figure out when we have 30
31:01so 30 is equal to 60 times 5/6 to the x/3 so you divide by 60 and you get 1.2 1/2 thats the half life is 5/6 to the x/3, now you want to figure out what x is we need logarithms this is why we have logarithms
31:39so far so good?
okay im gonna erase this so i have 1/2 is 5/6 to the x/3 so i take the log of 1/2 i can do any log i want
32:03so basically you take the log of both sides you take the square root of both sides and square both sides take the log of both sides you like logarithms so now what do you do?
make up a power in front thats why logs are so great
32:31log of 1/2 is x/3 log of 5/6 now you can isolate x that looks scary but remember log 5/6 just a number divided by a half is just some number this is what they use in chemistry by the way so what can you do? well you can multiply both sides by 3 so you get 3 times the log of 1/2 is x times the log of 5/6
33:03you divide by log 5/6 you get 3 log 1/2 is log 5/6 equals x you can certainly take log base 5/6 you can do log base 5/6 but thats kind of painful
33:31you could, you can also do the natural log of both sides you can do the log base whatever you want to do of both sides who was able to get to the end?
not many of you, good we have to do another one then got a question?
lets do one more then, ill make sure everyone gets this i got to do one more thing dont erase?
34:13i think we have a webassign due this week it wasnt posted yet?
ill have to talk to professor sutherland
34:54i hope we have review in before the weekend, exams before the weekend
35:39by the way you could use the half life formula but this is just as good, it gets you in the exact same spot
36:34if you have 20 ounces of an element 8 hours later you have 16 ounces what is the half life?
the what i kind of erased there so at time 0 we have 20 and at time 8 were down to 16 so time x your gonna need to have 10
37:02so we know y=a times b to the x and you know at 0 we get a so this is y=20 b to the x now i need to figure out the other part i plug in 8 for x and i get 16 for y so i ahd 16 is 20 to the 8
37:3216/20 is 4/5 or you can leave it 16/20 b is 4/5 to the 1/8 so far so good?
so y is 20 times 4/5 to the x/8 doing okay?
38:01doing alright? getting good grades?
i need to know wheni get 10 and we gonna divide 10 by 20 and get 1/2, were trying to find the half life
38:33so this is 4/5 to the x/8 when you have an equation and you have x in the power so you dont know the power thats when you use logarithms you take the log of both sides and now thanks to the power of logarithms
39:00you can bring the x over, right in front and you get log of 1/2 is x/8 log of 4/5 now multiply both sides by 8 and divide by log 4/5
39:35and you leave it like that, in fact you should leave it like that because you dont know what log 1/2 divided by log 4/5 is you put that in a calculator so far so good?
where does the 8 come from?
40:00we have x/8 thats in the denominator so you have x times the log 4/5 divided by 8 so you multiply both sides by 8 not you just have x times log 4/5 now the log of this is just a decimal thats just a number log of 1/2 is just a number, so you have 8 times some number divided by some other number if you were doing this with a calculator you could figure these out first you can do 8 times log of 1/2 equals x times something, divided by something
40:31but since you dont have a caslcultaor you have to get it down to that point lets do a couple other things with logarithms so0 remeber the log laws?
right them down, i think i put them up.. i posted them theres3 of them
41:02the log of a times b equals log a plus log b you have log of a divided by b log a minus log b and log of a to the b, this is the one we just used b times
41:30the log of a thats the most important one for those kind of equations we just did that says that if you have log of something raised to a power you can bring the power in front so what if i said i wanted you just to take something like
42:01i wanted to break that apart, remember using logarithms in high scool?
i have the log of x squared y cubed, thats the two things multiplied together so i can make that a log of x squared plus the log of y cubed this is very useful in calculus when you do something thats called logarithm differentiation very fancy word for something thats actually very helpful
42:39now i can break this up even more because log x squared is 2 times the log of x and log of y cubed is 3 times the log of y what if i wanted to break up
43:10log x to the 3 over y to the 5 well log x to the 3 over y to the 5 is log x to the 3
43:31minus log y to the 5 and i can take the powers and put them in front i can say this is 3 log x minus 5 log y so this is breaking up a log its very useful because you dont as much trouble as working in this form as you do working in this form first you have to be spectacle of both directions
44:03i give you some time to copy this down before i erase the board
44:39what if i wanted to break that up?
lets break this up we can make this log of x to the 4 square root of y minus log of z squared\ so far so good so log x to the 4 times the square root of y
45:01is two things multiplied together so i can make that lof of x to the 4 plus log square root of y minus log of z squared now i can put the powers in front 4 log x square root of y is the same as what power?
1/2 thats a 1/2 log y
45:32minus 2 log z howd you do on this one?
so far so good?
now were gonna go the other way ill wait till everybody copied this down square root of y is the 1/2 power
46:00so its the same as saying y to the 1/2 so thats why its 1/2 log y im losing a lot of you guys hang in there you only have a few more minutes okay so im gonna cover this up
46:40okay right that as a single log this becomes the log of x cubed this becomes the log of y squared and that becomes the log of z to the 4
47:03now we have log of x cubed plus log of y squared thats the log of x cubed times log squared and i got minus z to the 4 so z to the 4 goes in the denominator one last one make sure you guys get the concept what if i said
47:31i wrote what if i had that this is the one that tends to mess people up but t shouldnt 4 log x is still log x to the 4 right?
48:012 log y, is still log y squared 3 log z is still log z cubed this becomes log of, heres where heres where we mess it up
48:32thats where everybody messes up just because i have this one minus this one plus this one, doesnt make it any different that it would be this way so basically everything that i am adding goes into the numerator and everything that im subtracting goes into the denominator doesnt matter where i put them right?
49:00a minus b plus c is the same thing a a plus c minus b