Stony Brook MAT 123 Fall 2015
Lecture 27: more equation solving; review
November 30, 2015

Start   suppose theres something that looks like this if you see this the first thing you should know is e to the 2x is what what you have when you have x and square it if e to the x squared of x dont confuse that a e to the x squared
0:31anyway so thius.. what you can do is you can look at this and say see how that power is twice as big as that power?
its kind of like one of those quadratic things if you love some of them so why dont i do smething like this?
if i let y=e to the x i can rewrite this y squared -6y-7=0
1:00then i could factor this and i get y-7 times y+1 which means y=7 ir y=-1 how we doing so far?
you know this stuff? quadratics you canall do some little factor now you have to match, e to the x is 7
1:31or e to the x is -1 if e to the x is 7 x is the natural log of 7 and of e to the x i -1 now we get x is the natural log of -1 there is not natural log of -1 in real so this is no solution that make sense? e to the x always has to be something greater than 0
2:02can be 0 and cant be a negative how we doing on this so far?
make sure you understand of course
2:30you can have something like wanted to use similar numbers ' so e to the -x is 1/x to the positive x
3:03and then if you multiply 2 tot he x you get e to the 2x minus 8 equals 7 e to the x and see how i go that step i took this and multiplied 2 by e to the x so now i just subtract e to the x, 7e to the x from both sides
3:35so that looks like a quadratic no so to soslve that y= e to the x you get y squared minus 7y minus 8 equals 0 that factors
4:03y-8, y+1=0 y=8 y=-1 remember its naturally y natural e to the x e to the x is 8 or e to the x = -1 you have x is natural log of 8 or or does not exist ecause e to the x has to be greater then 0
4:34not equal to 0, greater then 0 everybody happy so far see something like that on webassign?
you didnt, you might now of course we dont have to use the function e to the x in it we can do another type of function
5:30what if i gave you something liek that well i could get y=sinx
6:08i would get y squared-3y+2=0 you get y-2 and y-1 equals 0 gand this is easy y=2 or y=1
6:30you can say its not actually y, its sinx this means sinx equals to sinx=1 and sinx cant equal 2 why cant sinx equal 2?
whats the biggest sinx can be?
1, right sinx is always between 1 and -1, because of the graph
7:01sinx can equal 1 and we all memorize this by now we know thats pi/2 or 90 degrees if youre doing the final and you can put it in degrees but you cant put it in radains put the right answer in degrees and get yourself most of the credit you should be able to do radains by now might get the right answer in degrees and then you out and cant get the radains
7:33could you say its the sin inverse of 1, umm you could its not 100 percent correct you should know the sinx 1 is pi/2 you would get some of the credit but you wouldnt get all of the credit now how can we make this more annoying?
if it said soolve for x, you would say x
8:05usually 0 to 2pi otherwise there is an infinity number of places for sinx equals 1 pi/2 5pi/2 9pi/2 -3pi/2 all sorts sinx is always between 1 and -1 think about the sin graph
8:34same with cosine, cosine is between 1 and -1 so far so good, alright now go back to this for a second, lets make it slightly more annoying what would you do if i gave you that?
9:01e to the 2x-6e to the x-5 you would do the same thing as before, do the substitution y=e to the x you get y squared -6y -5=0 and now you try and factor it into a facotr
9:31no factor, so what do you do?
you use the quadratic formula, the completing the square. why do you got to do that thoug?
thats why we invented the quadratic formula thats painful but this is 6 plus or minus the square root of 56 divided by 2 you dont need to divide
10:00so that means that e to the x either equals 6 plus the square root of 56/2 or 6 minus the square root of 56/2 now if your savey the square root of 56 is a little bigger then 7 7 squared is 49 so this will come out negative if you didnt get that, i would understand you dont have calculators
10:31unless its really screamingly obvious like i said 1-10 but its alright if you look at the exam at 6 minus the square root of 56/2 you dont need to streess at the exam and not realize its a negative number but anyways, x is the natural log og 6+ square root of 56/2 so its possible to have something like this and use the quadratic formula
11:02little trails with that but you know you got the exam youre not concentrating a b+ instead of minus and you look at the exam and say why am i having trouble with number 4 and youre like oh and then later people say oh that sucks
11:33we understand these?
one more practice how bout something like this
12:27i gave you all sorts of things
12:30ln -2 -5 will that make you happy?
sure alright how would we solve that?
13:07so what do we do on this? well do you remember the rules of logs?
the log of x-5 plus log of x-2 thats the same as log of x-5 times x-2 and of course, ln whatever you want to call it
13:30is log base e so thats the same as saying x-5 times x-2 equals e to the 0 and e to the 0 is 1 those are the rules of logs, so you get the log of something plus the log of something thats the same as the single log multiplied together remember natural log is log base e
14:00so that means e to the 0 equals whatever this is and e to the 0 is 1 so now we multiply this out and we get x squared minus 7x plus 10 equals 1 or x squared minus 7x plus 9 equals 0
14:43so now we factor that but it doesnt factor so you can sing the song when youre doing it
15:10so thats are either of those negative 7+ or minus the square root of 13 /2 is negative?
15:40no niether of those are negative the square root of 13 is not bigger then 7 7+the square root of 13/2 is positive 7- the square root of 13/2 is positive. they both are positive now thats thats it, your only problem is
16:02this happened in webassign 7+the square root of 13/2 minus 5 is positive minus 2 is positive but 7- the square root of 13/2 minus 5 is negative but somehow is expecting that answer anyway put both answer if you were to see something like this but you might not
16:30i would write that as seperate answers i would write 7+the square root of 13/2 and 7=the square root of 13/2 thats just me so this is the equation type of thing you should solve have you practice a couple?
i have exacrtly one more so lets make up a couple of these just to make sure you know this
17:06try to aviod the quadratic formula, you can do it. your call
18:27heres two to get you started
18:33so what do we do with this? multiply 2 to the x and we get e to the 2x-6=5eto the x or e to the 2x -5e to the x -6 equals 0 so far so good you can factor it out and make any substitutions
19:01or not it doesnt really matter but i like to do substitution to keep myself now we got y squared -5y - =0 and that factors quit nicely y-6 and y+1
19:31so far so good?
are we enjoying ourselves?
that means y=6 or y=-1 i did this one before e to the x equals 6 e to the x equals negative 1 x equals natural log of 6 x does not exist howd we do on this one?
20:01nice and easy we love this one now we do one with log what about this?
wel you can factor out sinx equals 4sin squared x-3 equals 0
20:31why would i do that?
i know why because now sinx is 0 or this is 0 and that becomes sin squared x equals 3/4
21:02qhich means you take the square root and you get sinx plus or minus the square root of 3/2 i said plus or minus didnt i did you remember the plus or minus?
x is 0 equals pi
21:33and this means x is ppi/3 2pi/3 4pi/3 or 5pi/3 of course if you put all of them you would get partial but not full credit if you dont get some of the you get some of the all credit
22:00and of course you try and do it in radians enough of this lets do some other stuff how we doing so far?
22:46can you solve that so if i subtract two logs thats the same as a single log divided by
23:09so far we all go that first step?
we all got partial credit?
and that means.. tats a 4 if you cant tell 4 to the 1 equals this
23:304 raised o the 1 which is 4 then you cross multiply and you get 2x plus 3 equals 4 times 3x plus 1 excited about that?
okay distribute 2x plus 3 equals 12x+4
24:07and then you get -1 is 1-x x is =1/10 can x be -1/10 it can be negative x can be negative this cant be negative that answer can come out negative as long as your are taking the log of pi/1
24:34when you plug in -1/10 they both come out positive thats okay thats your answer howd we do on that one?
not so much
25:19leys see if you can find the inverse lets practice finding the inverse, i suspect this would be an issue so first we replace f of x with y
25:35and we switch x and y now we have to isolate y so we divide both sides by 4 and take the natural log of both sides
26:04the natural log of x/4 equals 2y-1 add 1 and divide by 2
26:33that is the inverse function can we practice another one?
want another with e in it?
want a log?
make sure we got this?
27:49alright find the inverse function of x
28:02so we can make this y of 8 and 1+x sqitch variables and x is 8 thats the easy step gets you somewhere but not a lot what do we do with that 8 do the log base 8 of both sides or you can any log of both sides
28:31either way which is just easiest now you subtract log and divide by 3 you get log base 8 x minus 1
29:03we all see what i did with that one?
alright how bout this?
y is 5 log base 4 2x-3 switch x and y
29:38divide by 5 and then you do 4 to the x-5 2y-3 and you get 3
30:09and you get 4x/5 plus 3 over 2 equals y howd we do?
now if you have wanted to on this one
30:36when youre at the stage x=8 1+3y you couldve taken the log of both sides and do any log you want so for example natural log couldve done this and now you put the 1+3y in front
31:05aand you divide and you take it from there subtract one and dive by 3, you couldve done that this is a little smoother but both work put it in the calculator you get the same answer
31:33lets practice a couple more of these things
32:04okay find the inverse of that so you just switch x and y and then we just have to isolate y so we divide by 5
32:36and we cube bot sides and you get x/5 cbed can also be written as x cubed/5 cubed equals 4y-1 add 1 and divide by 4
33:10how do we feel about invereses?
we love inverse?
34:07jeres an entertaining one x+h wel f of x is 5x squared and f of x plus h
34:30is 5 x plus h squared minus 3 to the x+h so what you do is fill in both of the x's in x+h x squared plus h have to write x+h squared times x+h now we should simplify this before we do the subtraction
35:00this is 5 times x squared plus 2xh plus h squared -3x-3h then distribute you get 5x squared plus 10xh plus 5h squared minus 3x minus 3h now if you want to find f of x+h
35:33minus f of x this minus this 5x squared plus 10 xh plus 5x squared minus 3x minus 3h minus ex squared minus 3x you have to distribute that minus sign
36:02so thats 5x squared plus 10xh plus 5h squaed 3 to the x minus 3h minus 5x squared plus 3x and the 5x squared and the 3x's cancel and you get 10xh plus 5h squared minus 3h