|Start||normally so far when you see something you have x to something now were gonna have something to the x the base is a number thats a function with the form y equals b to the x, b is a number so for example you have y equals 2 to the x that means its something doubling|
|0:37||so to the is 1 because if you saw las time or the time before when i did exponentail functions anything to 0 is 1 2 to the 1 is 2 2 squared is 4, 2 cubed is 8 so everytime we go up another x by x goes up by 1 y doubles this is pretty straight forward reight so this function is doubling|
|1:04||now what happens if this is negative 1 another rule of exponents to the negative one is 1 over 2 to 1, so its a half and thats of negative 2 is 1/2 squared is 1/4 x is negative 3 is 1/2 cubed which is 1/8 and so on. notice i dont have any negative values here|
|1:33||when i plug in 0 positive numbers i get positive values i plug in negative values i also get positive values, because remember negative power means 1 over positive power if i did a root, so if i did a cubed of a 1/2 that just means the squared root of 2, thats still positive and 2 to the third, is cubed root of 2 still positive, so youre never gonna get a negative number now if i graph this|
|2:00||it would end up looking something like that that says 3,8 if you cant read my hand writting so notice what happens, as after x increases by 1 each time, the thing keeps going up faster and faster because its doubling|
|2:32||when x=10 youre gonna be at 1024
youre gonna shoot up in the x direction
in y direction, and as x gets to be more and more negative
youre just gonna approch the x axis
so this function has a horizontal asymptote
of the x axis okay?
theres no asymptote in this direction and it goes up pretty fast what do i mean by it goes up fast, compared to x sqaured okay if you have y=x squared
|3:02||that goes to point 10, 100, x=10, y=100 if you get y=2x x=10, y=1024 okay so it goes up much faster when x is when x gets to be a really be number like100 thats 10000, 2 to a hundred is a very big number|
|3:31||its lots of density exponential functions grow very quickly and go down very quickly thats when we talk about exponential growth and exponentail decay which you see i told you last time, you see in biology you see in physics, chemistry we do reaction rates a lot of reaction rates, you should love those 1st order, 2nd order, 3rd order, you guys dont know this yet no 1st order reactions|
|4:00||2nd order, its coming now its a pie waiting for you i know how exciting it is now what if instead of the 2 to the x you had 3 to the x just got to erase this|
|4:30||how bout y= 3 to the x well again i have to make a table 3 to the 0 is 1 so you still get 1 3 to the 1 is 3, 3 to the 2 is 9 3 to the 3 is 27 -1 is 1/3, -2 is 1/9 -3, 1/27 so a negative means 1 over the positive|
|5:02||so far so good?
so now remember whats going on with the 2 x is 3 2 and 3 is 8 here x=3, 2 and 3 is 27 so 3 to the x, is going up faster than 2 to the x but when x is negative 3 y=1/8 here when x=-3, y=27 which is smaller but also goes down faster so if wanted to compare
|5:35||that could be 2 to the x 3 to the x could go up faster and flatten faster notice they both go through 0,1 because anything that is 0 is 1 so you pick a point, x= whatever|
|6:01||3 to the x is bigger than 2 to the x
thats if x is greater tan 0
and pick some point here
3 to the x will be less than 2 to the x
that make sense?
why its gonna be bigger on one side and less than the other remember negative number means 1 over so 1 over a bigger number is 1 over a smaller number so thats what the exponential graphs look likw and thats what exponential growth is about
|6:30||theres a lot of things that grow exponentially, the cells in the body so you have the zygote thats the first one starts dividing. i havent tekne biology in such a long time in just 9 months you get a baby you have a lot of cells in a very small period of time things that grow exponentially, bacteria you have an infection and now all of a sound you have a hundred million of these things running around so exponential growth will be really quick|
|7:02||or it can come down really quickly a quadratic, cubic those slopes are much slower linear doesnt go very fast at all so what i look at is an exponential kind of growth problem|
|7:43||let me rephrase that sorry|
|8:38||typical word problem
so you start off
it will be some dish, a pastry dish or whatever
and you count the number of bacteria and you get 100
nice convient number
you come back 2 hours later and now theres 200
how may do you have after 24 hours?
|9:01||this is being doubled in 4 hours so its gonna double again and double again time 0, you have 200 time 4 hours you have 400 so in another 4 hours it will double again|
|9:37||going pretty quickly so at 24 hours your at 12,800 bacteria you see how i did that? i doubled it every 4 hours thats easier thats why you work in pencil folks|
|10:03||yes by the way if you work backwards 400, the powers gonna be 100 when you get old im not old thats right, thats why youre getting an A so every 4 hours the thing doubles now we can do this mathematically of course, what we do is we say|
|10:47||you can say y equals a times b to the x where a and b are constants you figure out thats what a general exponential equation looks like|
|11:04||you say time 0, or 200. so 200 a times b to the 0 and anything to 0 is 1 so a is 200 i crossed that out because b to the 0 is 1 cause anything to the 0 is 1 except for 0, you just dont worry about|
|11:30||so now i know i can rewrite this equation 200 times b to the x so now i know that four hours later theres 400 so 400 is 200 times b to the 4|
|12:01||so i divide by 200, so i get 2 is b to the 4th b to the 4 is certainly annoying because now im gonna have to find b to the 4th root of 2 so whats another way i can think of this so i dont have to do the fourth root of 2 well i could pick a 4 hour units so i can take 1 set of 4 hour units, and then try and get the at the end remember multiply it by 4 i can do that, or i can find the 4th root of 2. its not that hard you solve that and you get b=2 to the 4th|
|12:32||thats my equation y is 200 times 2 to the 1/4 to the x which is 200 times 2 to the x over 4 1/4 times x is x/4. that says 1/4 so this is another way to think of the four hour units|
|13:01||every time this goes up by another 4 the whole thing will go up another 1 so now if i want to find out how many there are after 24 hours i go my equation and i could say 200 y is 200, time 2 to the 24 over 4 24/4 is 6. so another words i have 6 sets of 4 hours so its double 6 times 1 2 3 4 5 6|
200 times 2 to the 6
which is 200
which is 12000
so far so good?
nice easy example without a calculator easier for me than you guys now lets make up a harder one you understand what were doing so far?
|14:04||lets go through the whole thing again so i have 200 bacteria sitting fish initial. so the time is 0 i have 200 bacteria so i say look ab equation, is gonna be a times b to the x so if times 0 thats initially, when they first start the clock i have 200 bacteria present so i have 200, a times be to the 0|
|14:32||if b to the 0 is 1, i can find out what a is
that means my equation, i can now get rid of the a
and replace with 200
everybody understand that step?
so whatcha gonna have to do for these equations were gonna have 2 sets of information the first set of information will help solve either a or b, usually a
|15:00||the second set of information will help you find b well first i say alright when x is 0 i know y is a times b to the 0 is 1, so i know y=1 im sorry its a times 1 so y comes out a i get the a value now i have y equals 200 times b to the x thats my first piece of information now i say alright i also know that after 4 hours i have 400 bacteria so 400 is 200 times be to the 4|
|15:32||because x is how long its been in there and y is how much i have so i divide each side by 200 so i get 2 is b to the 4 or b is 2 to the 4 its just a number you dont really care what id does i go to my equation and plug in my information. so 200 times 2 to the 4 to x which i could simplify to x/4 because i multiply those powers rules of exponents|
|16:01||you have a or b is raised to a power you multiply the power now i want to find out what i get at 24 hours, i just plug in 24 and solve well if b to the 4 is 2 you take the 4th root of both sides that says b is 2 to the 1/4 thats the way you can 4th root it|
|16:33||you raise that to the 4th you get 2 you raisse that to the 4th you get 2 so lets do another one of these to make sure you get it|
|17:00||youll get this dont worry|
|17:33||so do this, just changed the numbers, same basic idea okay there initially a thousand bacteria in a dish 3 hours later there are 5 thousand bacteria how many will there be after 24 hours, same concept now we have 3 hours and its going up by a factor of 5 its a little messier|
|18:01||what you do is i have y=a times b to the x so my first piece of information there is initially 1000 bacteria that basically says im going through the point 0,1000 0 comma 1000 so when x is 0, i have 1000 bacteria and when x is 3 i have 5000 bacteria so when x is 24|
|18:31||how many bacteria do i have really what im trying to do this isn going up in a straight line, its not linear its exponential so i dont use my y equals ax+b, i use a times b to the x so i take the first piece of information and get o,1000 and i could solve for a because when x=0, y=1000|
|19:00||y is 1000 a times, 1000 equals a times b to the 0 you can do 0=1 so 1000=a so y i can now say is 1000 times b to the x now i know when x=3, y=5000|
|19:32||so 5000 equals 1000 times be to the 3 divided both side by 1000 and i get 5 is b cubed so the cubed root of 5 is b also known as 5 to the 1/3|
|20:03||thats what cubed roots are for when you know what something cubed is you take the cubed root and then you get the something now i can take my equation and say y is 1000 times 5 to the 1/3 to the x is 1000|
|20:30||times 5 over 3 x to the 3 you multiply those powers 1/3 times x is x/3, a third of x okay now i just have to do the last piece of information so now i know that y is 1000 times 5 to the x/3|
|21:01||this is your multiplying factor this is how fast its growing, its growing in 5s its divided by 3 so every time i have a 3 hour increment i need to multiply whatever i have by 5 so when x=3, 3/3 is 1 so i have to multiply it by 5 when x=6 6/3 is 2, i will multiply 5 by squared, so ill multiply it again thats what makes it grow exponential you say this with compound interest|
|21:32||so now what happens in 24 hours thats times 5 to the 24/3 which is 5 to the 8 i dont happen to know what 5 to the 8 is off the top of my head thats a big number lets see gotta ennd with 25 thats all i know|
|22:00||i certainly hope theyll be easier
or your answer will be
or your answer will be this
which is 390625000, i dont know why you guys didnt do that in your head
what youre in stony brook
maybe in chemistry
you get these calculators in chemistry? you do?
|22:36||it helps with the calculator|
|23:06||should we do one more of these? do you think we got the idea?
so notice what i keep doing okay i take my first piece of information and it helps me figure out a
|23:30||so i can go to my original equation and i can replace a then i can do the information again and figure out where b is so now i can go in my equation and replace a and b and ill get here now that i have the equation i can solve for any number now i can say how many bacteria in a half hour|
|24:04||alright lets have you guys try one on your own|
|24:52||lets see if you can do that one lets do this one|
|25:00||y=a times b to the x i know that initially there are 20 bacteria and 12 hours later there aare 40 bacteria, it is doubling in 4 hours well one thing you can do is say look how many doubles will i have after 48 doubles 4 times so double double double double and youre done to figure it out mathematically, you sorta have to figure out the formulas|
|25:31||because if we had calculators wed say how many do we have after 37 hours or some annoying number so initially theres 20, 12 hours theres 40 how many will i have after 48 hours thats a question mark if you cant tell so lets figure that out so i say this 20 and b is 0 b to 0 is 1|
i go in my original equation and replace the a
and say y is 20
times b to the x
so far so good?
now when x=12, im gonna get 40 so 40=20 times b to the 12
|26:30||divide both sides by 20 and then you get 2 b to the 12 so i just do the 12th root which is the same as saying 2 to the 1/12 so now i can take this equation and replace the b and i can say y=20 times b to the 12th to the x or 20|
|27:00||b to the x over 12 dont multiply the 20 times the 2 now i just need to find out how many there are after 48 hours so its 2 to the 48/12 48/12 is 4|
|27:34||you know what 2 to the 4th is??
16 so this comes out 320. we might have you do that level of arithmetic on a test you can expect to figure out what 2 to the 4th is and multiply that by 20 not a hard number to do, or you leave the answer like this you know when is not simple down
|28:00||thats the basic idea behind exponential growth
i mean generally we want you to go all the way to the answer
if you dont go all the way there and you just dont do the last calculations
that should still be worth full credit but you know it might cost you a point
what happens is it may cost you a point and complain you get the point back but sometimes you dont
you know it depends is it tuesday?
basically were not interested on you doing the calculations but showing that you can do it
|28:35||how its set up okay, thats what really matters thats the basic idea between the exponential equations theres another kind of exponential equation i ask you to do what if i give you something like that|
|29:01||that way you can figure what x is|
|29:43||so 25 the problem is 5 to the x plus 3, we know thats equal 25 to something but theyre not the same base how would i solve that? well id tried and make them the same baseed i know that 25 is 5 squared|
|30:09||right 25 is 5 squared?
and now i can multiply these powers together and say 5 to the x plus 3 is 5 to the 2x minus 2
|30:30||alright i didnt touch the left side
what i did was look at the right side and say that 25 is the same
as 5 squared
and then i take these three powers, cause i have something to a power raised to a power
and multiply them together and now in order for this to be true
messed that up
makes sense okay?
|31:00||we saw that and we get x=5
nice and simple right?
make sure we got the concept down do it again make sure everyone has this before i erase the answer is just 5 x is 5 we go the idea, lets do another one
|31:51||so how would i do this?
i look and i say well 4 to the something and 8 to the something its not gonna work cant let them equal each other but i could let 4 be 2 squared
|32:07||and let 8 be 2 cubed now i multiply it by the powers and i det 2 to the 6th x minus 2 2 to the 3x plus 3. do you see where theses numbers are coming from, youre just multiplying|
|32:30||now i know 6x-2 has to equal 3x+3 so do a little algebra x=5/3 load up on your algebra folks the major reason people have trouble with calculus is not that they cant figure out whats going on but is the algebra|
|33:07||okay have you guys do one real fast and have you move on from here to the more important content we understand this one|
|33:34||that is a 9 the 9 is 3 squared 27 is 3 cubed|
|34:13||now i know 9 is 3 squared and 27 is 3 cubed so i can multiply the powers, so i get 3 to the 2x minus 4 is 3 to the 9 minus 3x|
|34:32||that means 2x-4
must equal 9-3x
so i get 5x
x is 13/5
howd we do on that?
we get 13/5?
we have a feeling of satisfaction okay lets move on to something harder suppose to know them for logarithms
|35:00||yes such happy voices
alright ready to learn about logarithms?
its actually very simple
|35:58||alright so 10 to 1 is 10|
|36:00||and 10 to the 2 is 100, right? we know this so it must be power of 10 that gives you 50 theres got to be a power of 10 that gives you 50. you just dont know what that power is but we know somewhere betwen one and 2 right so here is a power of 10 that will give us 50 and its 1 point something so we call that power the logarithm of 50|
|36:31||i know but heres the problem i know that 2 to the 5 is 32 and 2 to the 6 is 64 so there must be a power of 2 that gives you 50 so we can call that the log of 50 but that cant be the same log, they cant be the same number right because this is a number somewhere between 1 and 2|
|37:01||and this is number somewhere between 5 and 6 so how do we know the difference, so what we do we gonna write a number down here to remind us thats called the base of the logarithm so if i know 4 to the 2 is 16 and 4 to the 3 is 64 then theres a power of 4 that gives me 50 so thats also the log of 50 so dont get confused|
|37:31||i put a little 4 so thats the based of the logarithm now i can sort of generalize lets think about whats going on what is that x? that x is some number with a decimal usually and if i take the base take the little number here and raise it to x ill get this|
|38:01||so if i take 4, and raise it to this x ill get 50 and if i take 10 and raise this x ill get 50. theyre not the same x because they cant be they have to have different values this side has to be between 5 and 6 that value has to be between 1 and 2 this value has to be between 2 and 3 everyone understand so far> ill repeat if you need me to or you can rewind the video and watch|
|38:31||okay so in general if i know x=log base b of a that means if i take b and i raise it to the x i get a so a logarithm and exponential are related to each other in fact theyre inverses|
|39:00||if i put in a i get x and if i put x ill get a
so far so good?
so lets try an extent to a few of these i have to find the log base 5 of 25 we cant ask you to find complicated numbers, you need a calculator tables or something you have no idea how to find a log, just by guessing
|39:35||so 5 raised to some power has to give you 25, this means i take 5 and raise to the x ill get 25 x must be 2 because 5 squared is 25 if i had a log base 5 of 125 equals x that means if i take 5 and raise it to this x|
|40:03||i get 125
so x must be 3
how are we doing on thses so far?
you shoulve seen these before if you knew this well enough of course you wouldnt be in this class only the night befor
|40:32||placement test alright how bout log base 2 of 16 that says if i take 2 and raise it to the x i get 16 so x must be 4|
|41:01||get the point home?
well if i say log base 3 of x now im going the other way now im saying 3 to the 5
|41:33||so that means x=243 notice i can go both directions, i can know what base number is and find the x or i go look for the x and find basically the logarithm, the power do either of those|
|42:10||so now how am i going to use the logs
we know how to solve 5 x to the 25
we said x=2
what if i said 5 to the x is 20
how would i solve that?
|42:30||thats what logarithms are for logarithms were invented to solve these problems because here i can just say well this is 2 because 5 squared is 25 so here i have no idea what it is, its less than 2 i dont know exactly what it is in fact its log base 5 20 so were gonna practice converting|
|43:01||thats the answer i dont care what the actual number is thats one point 88 or something who knows whip out your calculator and find out when we give you these problems, this would be a perfectly good answer we wont expect you to put it so if i have 7 to the x is 100|
|43:30||x would be log base 7, 100 you get and idea how big it is because 7 squared is 49 so its got to be bigger than 2 7 cubed is less than 343 so its got to be less than 3, so it has to be somewhere between 2 and 3 so far so good this is how you can use a log to solve various simple exponential equations so lets figure out some stuff we can do with logs|
|44:17||so what is the log of 1/2 of b
how do you know?
log base b of 1 is 0 because anything to the 0 is 1
if i take b and raise it to the 0 ill get 1, remember thats what the equation tells us rught
b to the 0 is 1 well
anything to the 0 is 1
so log of 1 is always going to be 0
remember this for calculus okay?
log of 1 is 0 doesnt matter what the base is
|45:00||whats the log base of b 1 why is it 1? well b to the 1 b how bout log base b b raised to x thats gonna come out x|
|45:31||because thats really just saying i take b, what power do i raise it to to get b to the x i raise it to the x this is how you know these are inverses that equation helps you figure out what there inverse okay you also need to know the graph not gonna worry about it exactly when i raise something to the power|
|46:00||and i take a base and raise it to a different power i add the powers remember that so the logarithms are powers so if i had the log of ab we multiply together and we just add the logs so thats a very important rule, thats called a log law and thats law number one|
|46:32||theres 3 of them if i draw a box around it its probably important maybe not what if i had a log of a over b well thats gonna be the log of a minus the log of b|
|47:13||just like when you multiply you add the logs, and when you divide you subtract the logs even if i have the log of a raised to the b thats gonna be b times log of a because|
|47:31||im gonna multiply it by b you guys should have seen this back when you first started logs ill raise it up then so these are very useful to learn especially that last 1 suppose|
|48:03||i have 3 to the x equals 40
now i know that says the log
that x is log base 3 of 40
whats another way you can do this?
you know how you can square both sides and square root both sides and things of both sides, you can take the log of both sides you should see logs a lot in chemistry log scales, acid base
|48:30||what are you guys doing in chemisrty?
i take the log of both sides but i put the x in front because the log of a to the b
|49:01||is b times log of a so log of 3 to the x is x times' log of 3 and remember log of 3 is just a number right so some number with a decimal and thats just some number so x is log 40 over log 3 so if this were a test question that would be what we were looking for if you say 3 to the x is 40, whats x|
|49:32||log of 40 over log of 3 or you could of said like base 3 40 theyre equivalent so again lets say i had 7 to the x is 20 take the log of both sides|
|50:02||i can multiply the x in front and get x log 7 with the log of 20 if i divide log 20 by the log of 7|
|50:47||if i take the power and multiply the front multiply the bottom i take the power put it in front and multiply by log thats all|
|51:00||not the same as some of these other equations so lets move on to one slightly more messier what if i had that|
|51:35||i can take the log of both sides and i put the x+3 in fornt dont forget the parenthesis|
|52:02||okay now lets figure out what x is so i can divide by log 6 and subtract 3 and get x+3 log of 35 over log of 6 so x is log of 35 over log of 6 minus 3 you can say its tricky|
|52:31||you have to learn the simple work before you learn the hard ones, you understand that?
the log of 55 over the log of 6 is not the same as the log of 35/6 becareful doing that on a test or webassign the log of 35/6
|53:01||is the log of 35-6 dont confuse these two theres another rule that im gonna show you guys next times which will to show you what to do with one logarithm and the other so see everybody on monday|