WEBVTT
Kind: captions
Language: en
00:00:07.940 --> 00:00:12.940
normally so far when you see something you have x to something now were gonna have something to the x
00:00:13.860 --> 00:00:15.140
the base is a number
00:00:15.960 --> 00:00:18.520
thats a function with the form y equals
00:00:19.360 --> 00:00:20.800
b to the x, b is a number
00:00:22.700 --> 00:00:23.660
so for example
00:00:26.820 --> 00:00:27.940
you have y equals
00:00:28.620 --> 00:00:29.180
2 to the x
00:00:29.720 --> 00:00:31.420
that means its something doubling
00:00:37.980 --> 00:00:39.140
so to the is 1
00:00:40.580 --> 00:00:43.880
because if you saw las time or the time before when i did exponentail functions
00:00:44.160 --> 00:00:45.280
anything to 0 is 1
00:00:46.380 --> 00:00:47.180
2 to the 1 is 2
00:00:48.880 --> 00:00:50.560
2 squared is 4, 2 cubed is 8
00:00:51.340 --> 00:00:53.340
so everytime we go up another x
00:00:54.140 --> 00:00:55.100
by x goes up by 1
00:00:55.800 --> 00:00:56.440
y doubles
00:00:56.440 --> 00:00:57.880
this is pretty straight forward
00:00:58.340 --> 00:01:00.740
reight so this function is doubling
00:01:04.960 --> 00:01:07.440
now what happens if this is negative 1
00:01:07.940 --> 00:01:09.700
another rule of exponents
00:01:10.040 --> 00:01:11.320
to the negative one
00:01:11.760 --> 00:01:14.080
is 1 over 2 to 1, so its a half
00:01:16.200 --> 00:01:17.720
and thats of negative 2
00:01:18.760 --> 00:01:19.720
is 1/2 squared
00:01:20.220 --> 00:01:20.720
is 1/4
00:01:22.100 --> 00:01:23.060
x is negative 3
00:01:25.080 --> 00:01:25.880
is 1/2 cubed
00:01:28.100 --> 00:01:28.900
which is 1/8
00:01:28.900 --> 00:01:31.900
and so on. notice i dont have any negative values here
00:01:33.860 --> 00:01:34.820
when i plug in 0
00:01:34.820 --> 00:01:37.320
positive numbers i get positive values
00:01:37.320 --> 00:01:43.220
i plug in negative values i also get positive values, because remember negative power means 1 over positive power
00:01:44.360 --> 00:01:46.920
if i did a root, so if i did a cubed of a 1/2
00:01:47.360 --> 00:01:49.580
that just means the squared root of 2, thats still positive
00:01:49.820 --> 00:01:52.140
and 2 to the third, is cubed root of 2
00:01:52.140 --> 00:01:54.180
still positive, so youre never gonna get a negative number
00:01:57.100 --> 00:01:58.300
now if i graph this
00:02:00.600 --> 00:02:02.520
it would end up looking something like that
00:02:16.280 --> 00:02:18.440
that says 3,8 if you cant read my hand writting
00:02:24.560 --> 00:02:26.320
so notice what happens, as
00:02:26.320 --> 00:02:30.960
after x increases by 1 each time, the thing keeps going up faster and faster because its doubling
00:02:32.040 --> 00:02:35.340
when x=10 youre gonna be at 1024
00:02:35.640 --> 00:02:38.200
youre gonna shoot up in the x direction
00:02:38.940 --> 00:02:43.460
in y direction, and as x gets to be more and more negative
00:02:43.460 --> 00:02:45.640
youre just gonna approch the x axis
00:02:45.640 --> 00:02:48.400
so this function has a horizontal asymptote
00:02:49.100 --> 00:02:50.500
of the x axis okay?
00:02:51.520 --> 00:02:54.060
theres no asymptote in this direction and it goes up pretty fast
00:02:54.760 --> 00:02:58.360
what do i mean by it goes up fast, compared to x sqaured
00:02:59.580 --> 00:03:01.420
okay if you have y=x squared
00:03:02.140 --> 00:03:03.580
that goes to point 10,
00:03:04.100 --> 00:03:05.780
100, x=10, y=100
00:03:06.620 --> 00:03:07.580
if you get y=2x
00:03:13.660 --> 00:03:16.580
x=10, y=1024
00:03:17.680 --> 00:03:19.600
okay so it goes up much faster
00:03:22.680 --> 00:03:23.240
when x is
00:03:24.000 --> 00:03:26.800
when x gets to be a really be number like100
00:03:27.200 --> 00:03:30.240
thats 10000, 2 to a hundred is a very big number
00:03:31.500 --> 00:03:32.780
its lots of density
00:03:33.620 --> 00:03:35.220
exponential functions
00:03:36.080 --> 00:03:38.880
grow very quickly and go down very quickly
00:03:39.360 --> 00:03:43.840
thats when we talk about exponential growth and exponentail decay
00:03:43.840 --> 00:03:47.360
which you see i told you last time, you see in biology you see in physics, chemistry
00:03:48.520 --> 00:03:49.880
we do reaction rates
00:03:49.960 --> 00:03:53.000
a lot of reaction rates, you should love those
00:03:53.920 --> 00:03:56.500
1st order, 2nd order, 3rd order, you guys dont know this yet
00:03:58.920 --> 00:04:00.440
no 1st order reactions
00:04:00.560 --> 00:04:02.000
2nd order, its coming
00:04:02.840 --> 00:04:04.680
now its a pie waiting for you
00:04:05.580 --> 00:04:07.180
i know how exciting it is
00:04:10.240 --> 00:04:12.080
now what if instead of the 2 to the x
00:04:12.960 --> 00:04:14.000
you had 3 to the x
00:04:14.320 --> 00:04:15.760
just got to erase this
00:04:30.960 --> 00:04:32.240
how bout y= 3 to the x
00:04:32.660 --> 00:04:33.380
well again
00:04:34.000 --> 00:04:35.360
i have to make a table
00:04:37.040 --> 00:04:38.960
3 to the 0 is 1 so you still get 1
00:04:39.840 --> 00:04:42.520
3 to the 1 is 3, 3 to the 2 is 9
00:04:43.460 --> 00:04:44.880
3 to the 3 is 27
00:04:46.140 --> 00:04:49.100
-1 is 1/3, -2 is 1/9
00:04:50.260 --> 00:04:52.060
-3, 1/27
00:04:53.500 --> 00:04:56.060
so a negative means 1 over the positive
00:05:02.360 --> 00:05:03.320
so far so good?
00:05:03.320 --> 00:05:05.500
so now remember whats going on with the 2
00:05:05.920 --> 00:05:06.420
x is
00:05:07.700 --> 00:05:08.200
3
00:05:08.840 --> 00:05:10.020
2 and 3 is 8
00:05:10.580 --> 00:05:11.940
here x=3, 2 and 3 is 27
00:05:12.340 --> 00:05:15.220
so 3 to the x, is going up faster than 2 to the x
00:05:17.840 --> 00:05:19.360
but when x is negative 3
00:05:20.100 --> 00:05:20.600
y=1/8
00:05:20.740 --> 00:05:22.500
here when x=-3, y=27 which
00:05:22.900 --> 00:05:23.620
is smaller
00:05:24.060 --> 00:05:25.740
but also goes down faster
00:05:26.220 --> 00:05:27.740
so if wanted to compare
00:05:35.480 --> 00:05:36.920
that could be 2 to the x
00:05:38.600 --> 00:05:40.360
3 to the x could go up faster
00:05:42.060 --> 00:05:43.340
and flatten faster
00:05:48.060 --> 00:05:50.920
notice they both go through 0,1
00:05:51.140 --> 00:05:53.140
because anything that is 0 is 1
00:05:54.820 --> 00:05:58.380
so you pick a point, x= whatever
00:06:01.320 --> 00:06:03.400
3 to the x is bigger than 2 to the x
00:06:03.400 --> 00:06:04.760
thats if x is greater tan 0
00:06:05.360 --> 00:06:06.960
and pick some point here
00:06:09.360 --> 00:06:10.960
3 to the x will be less than 2 to the x
00:06:11.020 --> 00:06:12.140
that make sense?
00:06:12.140 --> 00:06:14.620
why its gonna be bigger on one side and less than the other
00:06:14.620 --> 00:06:16.660
remember negative number means 1 over
00:06:16.980 --> 00:06:20.260
so 1 over a bigger number is 1 over a smaller number
00:06:22.940 --> 00:06:26.320
so thats what the exponential graphs look likw
00:06:26.360 --> 00:06:27.920
and thats what exponential growth
00:06:28.660 --> 00:06:29.220
is about
00:06:30.600 --> 00:06:36.600
theres a lot of things that grow exponentially, the cells in the body so you have the zygote
00:06:36.980 --> 00:06:38.260
thats the first one
00:06:38.260 --> 00:06:41.960
starts dividing. i havent tekne biology in such a long time
00:06:43.880 --> 00:06:47.500
in just 9 months you get a baby
00:06:49.500 --> 00:06:52.140
you have a lot of cells in a very small period of time
00:06:52.920 --> 00:06:54.880
things that grow exponentially, bacteria
00:06:54.880 --> 00:06:59.580
you have an infection and now all of a sound you have a hundred million of these things
00:06:59.580 --> 00:07:02.460
running around so exponential growth will be really quick
00:07:02.460 --> 00:07:04.620
or it can come down really quickly
00:07:06.960 --> 00:07:10.160
a quadratic, cubic those slopes are much slower
00:07:10.160 --> 00:07:12.320
linear doesnt go very fast at all
00:07:14.100 --> 00:07:15.140
so what i look at
00:07:15.140 --> 00:07:17.600
is an exponential kind of growth problem
00:07:43.160 --> 00:07:44.920
let me rephrase that sorry
00:08:38.160 --> 00:08:39.040
typical word problem
00:08:46.020 --> 00:08:47.060
so you start off
00:08:47.060 --> 00:08:50.080
it will be some dish, a pastry dish or whatever
00:08:50.080 --> 00:08:53.440
and you count the number of bacteria and you get 100
00:08:53.660 --> 00:08:55.100
nice convient number
00:08:55.100 --> 00:08:57.560
you come back 2 hours later and now theres 200
00:08:57.560 --> 00:08:59.680
how may do you have after 24 hours?
00:09:01.280 --> 00:09:03.360
this is being doubled in 4 hours
00:09:03.360 --> 00:09:05.020
so its gonna double again
00:09:05.620 --> 00:09:06.740
and double again
00:09:08.840 --> 00:09:12.000
time 0, you have 200
00:09:13.920 --> 00:09:14.720
time 4 hours
00:09:15.080 --> 00:09:15.880
you have 400
00:09:17.260 --> 00:09:18.620
so in another 4 hours
00:09:18.720 --> 00:09:20.080
it will double again
00:09:37.380 --> 00:09:38.820
going pretty quickly
00:09:39.740 --> 00:09:43.380
so at 24 hours your at 12,800 bacteria
00:09:43.380 --> 00:09:46.160
you see how i did that? i doubled it every 4 hours
00:09:52.340 --> 00:09:53.220
thats easier
00:09:57.660 --> 00:09:59.900
thats why you work in pencil folks
00:10:03.760 --> 00:10:07.080
yes by the way if you work backwards 400, the powers gonna be 100
00:10:09.780 --> 00:10:10.820
when you get old
00:10:15.120 --> 00:10:18.480
im not old thats right, thats why youre getting an A
00:10:24.740 --> 00:10:26.980
so every 4 hours the thing doubles
00:10:29.020 --> 00:10:33.100
now we can do this mathematically of course, what we do is we say
00:10:47.920 --> 00:10:48.640
you can say y
00:10:49.460 --> 00:10:50.020
equals a
00:10:50.420 --> 00:10:53.240
times b to the x
00:10:54.740 --> 00:10:58.620
where a and b are constants you figure out
00:10:59.520 --> 00:11:03.120
thats what a general exponential equation looks like
00:11:04.900 --> 00:11:08.920
you say time 0, or 200. so 200
00:11:10.340 --> 00:11:12.520
a times b to the 0
00:11:14.720 --> 00:11:16.360
and anything to 0 is 1
00:11:18.660 --> 00:11:19.300
so a is 200
00:11:20.260 --> 00:11:22.820
i crossed that out because b to the 0 is 1
00:11:24.100 --> 00:11:25.860
cause anything to the 0 is 1
00:11:25.980 --> 00:11:28.540
except for 0, you just dont worry about
00:11:30.100 --> 00:11:32.740
so now i know i can rewrite this equation
00:11:33.620 --> 00:11:34.120
200
00:11:35.800 --> 00:11:36.760
times b to the x
00:11:47.180 --> 00:11:49.680
so now i know that four hours later
00:11:50.500 --> 00:11:51.220
theres 400
00:11:52.080 --> 00:11:54.900
so 400 is 200
00:11:57.140 --> 00:11:58.100
times b to the 4
00:12:01.140 --> 00:12:05.460
so i divide by 200, so i get 2 is b to the 4th
00:12:06.580 --> 00:12:10.440
b to the 4 is certainly annoying because now im gonna have to find b to the 4th root of 2
00:12:10.440 --> 00:12:14.300
so whats another way i can think of this so i dont have to do the fourth root of 2
00:12:14.960 --> 00:12:16.960
well i could pick a 4 hour units
00:12:17.220 --> 00:12:21.540
so i can take 1 set of 4 hour units, and then try and get the at the end
00:12:21.540 --> 00:12:23.160
remember multiply it by 4
00:12:23.160 --> 00:12:26.400
i can do that, or i can find the 4th root of 2. its not that hard
00:12:27.940 --> 00:12:32.300
you solve that and you get b=2 to the 4th
00:12:32.880 --> 00:12:34.080
thats my equation
00:12:34.620 --> 00:12:36.500
y is 200
00:12:37.640 --> 00:12:40.200
times 2 to the 1/4
00:12:41.300 --> 00:12:41.800
to the x
00:12:43.540 --> 00:12:44.340
which is 200
00:12:45.360 --> 00:12:48.860
times 2 to the x over 4
00:12:50.780 --> 00:12:54.540
1/4 times x is x/4. that says 1/4
00:12:57.780 --> 00:13:01.220
so this is another way to think of the four hour units
00:13:01.220 --> 00:13:03.420
every time this goes up by another 4
00:13:03.740 --> 00:13:06.060
the whole thing will go up another 1
00:13:06.060 --> 00:13:09.580
so now if i want to find out how many there are after 24 hours
00:13:11.020 --> 00:13:13.260
i go my equation and i could say 200
00:13:13.820 --> 00:13:16.880
y is 200, time 2 to the 24
00:13:18.340 --> 00:13:18.840
over 4
00:13:20.760 --> 00:13:22.520
24/4 is 6. so another words
00:13:22.760 --> 00:13:25.680
i have 6 sets of 4 hours so its double
00:13:26.040 --> 00:13:26.880
6 times
00:13:27.840 --> 00:13:29.440
1 2 3 4 5 6
00:13:31.740 --> 00:13:32.240
thats
00:13:33.780 --> 00:13:36.080
200 times 2 to the 6
00:13:37.280 --> 00:13:38.080
which is 200
00:13:38.820 --> 00:13:40.340
times 64
00:13:41.260 --> 00:13:42.220
which is 12000
00:13:43.640 --> 00:13:44.140
800
00:13:46.340 --> 00:13:47.300
so far so good?
00:13:47.720 --> 00:13:50.360
nice easy example without a calculator
00:13:51.340 --> 00:13:53.100
easier for me than you guys
00:13:56.180 --> 00:13:58.020
now lets make up a harder one
00:13:59.140 --> 00:14:00.480
you understand what were doing so far?
00:14:04.200 --> 00:14:05.620
lets go through the whole thing again
00:14:06.660 --> 00:14:08.980
so i have 200 bacteria sitting fish
00:14:10.780 --> 00:14:12.380
initial. so the time is 0
00:14:12.560 --> 00:14:13.840
i have 200 bacteria
00:14:13.920 --> 00:14:14.720
so i say look
00:14:15.560 --> 00:14:18.200
ab equation, is gonna be a times b to the x
00:14:19.300 --> 00:14:20.440
so if times 0
00:14:24.220 --> 00:14:26.480
thats initially, when they first start the clock
00:14:26.820 --> 00:14:28.660
i have 200 bacteria present
00:14:29.160 --> 00:14:31.160
so i have 200, a times be to the 0
00:14:32.380 --> 00:14:34.860
if b to the 0 is 1, i can find out what a is
00:14:35.380 --> 00:14:35.880
a=200
00:14:36.400 --> 00:14:39.180
that means my equation, i can now get rid of the a
00:14:40.280 --> 00:14:41.640
and replace with 200
00:14:43.700 --> 00:14:44.800
everybody understand that step?
00:14:48.560 --> 00:14:49.120
yes? no?
00:14:50.220 --> 00:14:53.340
so whatcha gonna have to do for these equations
00:14:53.620 --> 00:14:56.100
were gonna have 2 sets of information
00:14:56.100 --> 00:15:00.060
the first set of information will help solve either a or b, usually a
00:15:00.060 --> 00:15:02.720
the second set of information will help you find b
00:15:04.060 --> 00:15:06.640
well first i say alright when x is 0
00:15:07.340 --> 00:15:10.140
i know y is a times b to the 0
00:15:10.420 --> 00:15:11.960
is 1, so i know y=1
00:15:12.060 --> 00:15:13.420
im sorry its a times 1
00:15:14.640 --> 00:15:17.160
so y comes out a
00:15:17.380 --> 00:15:18.420
i get the a value
00:15:18.720 --> 00:15:21.120
now i have y equals 200 times b to the x
00:15:21.360 --> 00:15:23.760
thats my first piece of information
00:15:24.460 --> 00:15:27.180
now i say alright i also know that after 4 hours
00:15:27.460 --> 00:15:28.740
i have 400 bacteria
00:15:29.560 --> 00:15:32.140
so 400 is 200 times be to the 4
00:15:32.440 --> 00:15:34.920
because x is how long its been in there
00:15:35.200 --> 00:15:36.640
and y is how much i have
00:15:38.760 --> 00:15:43.460
so i divide each side by 200 so i get 2 is b to the 4 or b is 2 to the 4
00:15:44.320 --> 00:15:46.700
its just a number you dont really care what id does
00:15:48.240 --> 00:15:54.240
i go to my equation and plug in my information. so 200 times 2 to the 4 to x
00:15:54.460 --> 00:15:58.540
which i could simplify to x/4 because i multiply those powers
00:15:59.120 --> 00:16:00.400
rules of exponents
00:16:01.260 --> 00:16:04.940
you have a or b is raised to a power you multiply the power
00:16:06.580 --> 00:16:10.420
now i want to find out what i get at 24 hours, i just plug in 24 and solve
00:16:18.160 --> 00:16:20.860
well if b to the 4 is 2
00:16:21.380 --> 00:16:23.620
you take the 4th root of both sides
00:16:24.180 --> 00:16:27.600
that says b is 2 to the 1/4
00:16:27.600 --> 00:16:29.440
thats the way you can 4th root it
00:16:33.360 --> 00:16:37.760
you raise that to the 4th you get 2 you raisse that to the 4th you get 2
00:16:54.240 --> 00:16:55.880
so lets do another one of these to make sure you get it
00:17:00.700 --> 00:17:02.380
youll get this dont worry
00:17:33.120 --> 00:17:35.600
so do this, just changed the numbers, same basic idea
00:17:43.800 --> 00:17:46.580
okay there initially a thousand bacteria in a dish
00:17:48.400 --> 00:17:49.280
3 hours later
00:17:49.280 --> 00:17:51.220
there are 5 thousand bacteria
00:17:52.480 --> 00:17:55.000
how many will there be after 24 hours, same concept
00:17:55.860 --> 00:17:59.060
now we have 3 hours and its going up by a factor of 5
00:17:59.060 --> 00:17:59.580
its a little messier
00:18:01.660 --> 00:18:06.300
what you do is i have y=a times
00:18:07.300 --> 00:18:07.860
b to the x
00:18:10.800 --> 00:18:12.960
so my first piece of information
00:18:13.680 --> 00:18:15.920
there is initially 1000 bacteria
00:18:15.920 --> 00:18:19.400
that basically says im going through the point 0,1000
00:18:20.360 --> 00:18:21.160
0 comma 1000
00:18:22.400 --> 00:18:24.640
so when x is 0, i have 1000 bacteria
00:18:25.100 --> 00:18:25.980
and when x is 3
00:18:26.640 --> 00:18:28.000
i have 5000 bacteria
00:18:29.240 --> 00:18:30.780
so when x is 24
00:18:31.060 --> 00:18:32.820
how many bacteria do i have
00:18:32.820 --> 00:18:34.200
really what im trying to do
00:18:34.840 --> 00:18:38.200
this isn going up in a straight line, its not linear
00:18:38.280 --> 00:18:39.400
its exponential
00:18:41.620 --> 00:18:43.140
so i dont use my y equals
00:18:44.380 --> 00:18:48.040
ax+b, i use a times b to the x
00:18:48.580 --> 00:18:52.180
so i take the first piece of information and get o,1000
00:18:52.440 --> 00:18:53.880
and i could solve for a
00:18:54.660 --> 00:18:56.860
because when x=0, y=1000
00:19:00.940 --> 00:19:04.640
y is 1000 a times, 1000 equals a times b to the 0
00:19:07.820 --> 00:19:09.040
you can do 0=1
00:19:11.080 --> 00:19:12.580
so 1000=a
00:19:14.580 --> 00:19:16.780
so y i can now say is 1000
00:19:18.160 --> 00:19:19.120
times b to the x
00:19:27.380 --> 00:19:30.620
now i know when x=3, y=5000
00:19:32.400 --> 00:19:32.900
so 5000
00:19:35.300 --> 00:19:36.100
equals 1000
00:19:37.500 --> 00:19:39.240
times be to the 3
00:19:45.340 --> 00:19:47.020
divided both side by 1000
00:19:48.520 --> 00:19:49.160
and i get 5
00:19:50.260 --> 00:19:51.220
is b cubed
00:19:52.180 --> 00:19:53.300
so the cubed root
00:19:54.360 --> 00:19:54.860
of 5
00:19:56.600 --> 00:19:58.080
is b also known as
00:19:59.020 --> 00:20:01.240
5 to the 1/3
00:20:03.240 --> 00:20:07.720
thats what cubed roots are for when you know what something cubed is you take the cubed root and then you get the something
00:20:18.460 --> 00:20:21.700
now i can take my equation and say y is 1000
00:20:23.620 --> 00:20:25.640
times 5 to the 1/3
00:20:26.760 --> 00:20:29.040
to the x is 1000
00:20:30.660 --> 00:20:33.180
times 5 over 3 x to the 3
00:20:34.600 --> 00:20:36.360
you multiply those powers
00:20:38.180 --> 00:20:41.360
1/3 times x is x/3, a third of x
00:20:51.220 --> 00:20:53.720
okay now i just have to do the last piece of information
00:20:55.100 --> 00:20:56.780
so now i know that y is 1000
00:20:57.640 --> 00:20:58.140
times 5
00:20:59.480 --> 00:21:00.120
to the x/3
00:21:01.340 --> 00:21:06.540
this is your multiplying factor this is how fast its growing, its growing in 5s
00:21:07.720 --> 00:21:11.240
its divided by 3 so every time i have a 3 hour increment
00:21:12.080 --> 00:21:14.560
i need to multiply whatever i have by 5
00:21:15.400 --> 00:21:19.320
so when x=3, 3/3 is 1 so i have to multiply it by 5
00:21:19.940 --> 00:21:20.960
when x=6
00:21:21.960 --> 00:21:26.040
6/3 is 2, i will multiply 5 by squared, so ill multiply it again
00:21:26.040 --> 00:21:28.120
thats what makes it grow exponential
00:21:29.220 --> 00:21:31.620
you say this with compound interest
00:21:32.880 --> 00:21:35.840
so now what happens in 24 hours
00:21:37.020 --> 00:21:39.600
thats times 5 to the 24/3
00:21:43.020 --> 00:21:44.140
which is 5 to the 8
00:21:45.360 --> 00:21:47.840
i dont happen to know what 5 to the 8 is off the top of my head
00:21:48.660 --> 00:21:49.860
thats a big number
00:21:50.940 --> 00:21:51.500
lets see
00:21:53.820 --> 00:21:55.980
gotta ennd with 25 thats all i know
00:22:00.800 --> 00:22:02.180
i certainly hope theyll be easier
00:22:04.440 --> 00:22:05.880
or your answer will be
00:22:07.920 --> 00:22:09.680
or your answer will be this
00:22:11.840 --> 00:22:18.440
which is 390625000, i dont know why you guys didnt do that in your head
00:22:19.360 --> 00:22:21.040
what youre in stony brook
00:22:25.820 --> 00:22:27.100
maybe in chemistry
00:22:28.520 --> 00:22:31.720
you get these calculators in chemistry? you do?
00:22:36.000 --> 00:22:37.920
it helps with the calculator
00:23:06.540 --> 00:23:08.860
should we do one more of these? do you think we got the idea?
00:23:18.480 --> 00:23:20.560
so notice what i keep doing okay
00:23:26.260 --> 00:23:30.420
i take my first piece of information and it helps me figure out a
00:23:30.420 --> 00:23:32.320
so i can go to my original equation
00:23:32.580 --> 00:23:33.780
and i can replace a
00:23:34.520 --> 00:23:37.740
then i can do the information again and figure out where b is
00:23:37.740 --> 00:23:39.320
so now i can go in my equation
00:23:40.080 --> 00:23:43.700
and replace a and b and ill get here
00:23:44.960 --> 00:23:47.540
now that i have the equation i can solve for any number
00:23:47.680 --> 00:23:50.560
now i can say how many bacteria in a half hour
00:24:04.340 --> 00:24:06.000
alright lets have you guys try one on your own
00:24:52.360 --> 00:24:54.280
lets see if you can do that one
00:24:56.400 --> 00:24:57.440
lets do this one
00:25:00.760 --> 00:25:01.960
y=a times b to the x
00:25:06.760 --> 00:25:08.540
i know that initially
00:25:08.980 --> 00:25:10.420
there are 20 bacteria
00:25:11.320 --> 00:25:12.520
and 12 hours later
00:25:12.660 --> 00:25:15.860
there aare 40 bacteria, it is doubling in 4 hours
00:25:18.020 --> 00:25:23.780
well one thing you can do is say look how many doubles will i have after 48 doubles 4 times
00:25:24.460 --> 00:25:26.480
so double double double double and youre done
00:25:26.900 --> 00:25:30.300
to figure it out mathematically, you sorta have to figure out the formulas
00:25:31.200 --> 00:25:35.280
because if we had calculators wed say how many do we have after
00:25:35.680 --> 00:25:36.240
37 hours
00:25:36.240 --> 00:25:37.720
or some annoying number
00:25:39.500 --> 00:25:43.660
so initially theres 20, 12 hours theres 40
00:25:43.880 --> 00:25:47.060
how many will i have after 48 hours
00:25:47.060 --> 00:25:49.340
thats a question mark if you cant tell
00:25:51.680 --> 00:25:53.200
so lets figure that out
00:25:53.720 --> 00:25:55.360
so i say this 20
00:25:56.960 --> 00:25:57.780
and b is 0
00:25:59.520 --> 00:26:00.080
b to 0 is 1
00:26:02.360 --> 00:26:02.860
a=20
00:26:02.860 --> 00:26:05.720
i go in my original equation and replace the a
00:26:06.260 --> 00:26:07.460
and say y is 20
00:26:08.960 --> 00:26:11.400
times b to the x
00:26:12.560 --> 00:26:13.520
so far so good?
00:26:17.700 --> 00:26:21.000
now when x=12, im gonna get 40
00:26:21.540 --> 00:26:26.500
so 40=20 times b to the 12
00:26:30.880 --> 00:26:32.400
divide both sides by 20
00:26:33.500 --> 00:26:36.060
and then you get 2 b to the 12
00:26:37.340 --> 00:26:39.700
so i just do the 12th root
00:26:40.940 --> 00:26:43.500
which is the same as saying 2 to the 1/12
00:26:46.320 --> 00:26:49.440
so now i can take this equation and replace the b
00:26:50.600 --> 00:26:51.720
and i can say y=20
00:26:53.540 --> 00:26:54.740
times b to the 12th
00:26:56.980 --> 00:26:57.480
to the x
00:26:58.860 --> 00:26:59.360
or 20
00:27:00.360 --> 00:27:01.920
b to the x over 12
00:27:12.960 --> 00:27:15.040
dont multiply the 20 times the 2
00:27:16.460 --> 00:27:20.300
now i just need to find out how many there are after 48 hours
00:27:21.580 --> 00:27:23.560
so its 2 to the 48/12
00:27:27.740 --> 00:27:29.000
48/12 is 4
00:27:34.500 --> 00:27:36.420
you know what 2 to the 4th is??
00:27:37.420 --> 00:27:37.920
16
00:27:39.820 --> 00:27:44.780
so this comes out 320. we might have you do that level of arithmetic on a test
00:27:46.280 --> 00:27:48.680
you can expect to figure out what 2 to the 4th is
00:27:48.680 --> 00:27:50.160
and multiply that by 20
00:27:50.860 --> 00:27:54.540
not a hard number to do, or you leave the answer like this
00:27:56.680 --> 00:27:58.760
you know when is not simple down
00:28:00.620 --> 00:28:03.180
thats the basic idea behind exponential growth
00:28:09.820 --> 00:28:12.820
i mean generally we want you to go all the way to the answer
00:28:12.820 --> 00:28:17.020
if you dont go all the way there and you just dont do the last calculations
00:28:17.020 --> 00:28:20.980
that should still be worth full credit but you know it might cost you a point
00:28:21.720 --> 00:28:26.420
what happens is it may cost you a point and complain you get the point back but sometimes you dont
00:28:26.420 --> 00:28:28.600
you know it depends is it tuesday?
00:28:29.940 --> 00:28:34.720
basically were not interested on you doing the calculations but showing that you can do it
00:28:35.220 --> 00:28:37.780
how its set up okay, thats what really matters
00:28:41.580 --> 00:28:44.800
thats the basic idea between the exponential equations
00:28:44.800 --> 00:28:48.440
theres another kind of exponential equation i ask you to do
00:28:59.640 --> 00:29:01.740
what if i give you something like that
00:29:01.800 --> 00:29:03.880
that way you can figure what x is
00:29:43.940 --> 00:29:44.620
so 25
00:29:45.360 --> 00:29:49.680
the problem is 5 to the x plus 3, we know thats equal 25 to something
00:29:49.680 --> 00:29:51.320
but theyre not the same base
00:29:51.320 --> 00:29:54.620
how would i solve that? well id tried and make them the same baseed
00:29:56.000 --> 00:29:57.680
i know that 25 is 5 squared
00:30:09.060 --> 00:30:10.500
right 25 is 5 squared?
00:30:13.060 --> 00:30:14.500
and now i can multiply
00:30:14.900 --> 00:30:16.420
these powers together
00:30:17.280 --> 00:30:18.720
and say 5 to the x plus 3
00:30:19.860 --> 00:30:20.660
is 5 to the 2x
00:30:22.380 --> 00:30:22.880
minus 2
00:30:30.040 --> 00:30:32.360
alright i didnt touch the left side
00:30:32.360 --> 00:30:34.740
what i did was look at the right side and say that 25 is the same
00:30:34.940 --> 00:30:35.740
as 5 squared
00:30:36.440 --> 00:30:40.520
and then i take these three powers, cause i have something to a power raised to a power
00:30:41.500 --> 00:30:45.020
and multiply them together and now in order for this to be true
00:30:47.200 --> 00:30:48.160
messed that up
00:30:50.960 --> 00:30:52.660
x+3=2x-2
00:30:53.580 --> 00:30:54.780
makes sense okay?
00:31:00.760 --> 00:31:02.360
we saw that and we get x=5
00:31:05.640 --> 00:31:07.160
nice and simple right?
00:31:08.780 --> 00:31:10.140
make sure we got the concept down
00:31:12.040 --> 00:31:12.760
do it again
00:31:13.680 --> 00:31:16.480
make sure everyone has this before i erase
00:31:20.400 --> 00:31:21.680
the answer is just 5
00:31:22.740 --> 00:31:23.240
x is 5
00:31:25.260 --> 00:31:27.500
we go the idea, lets do another one
00:31:51.240 --> 00:31:52.680
so how would i do this?
00:31:53.520 --> 00:31:57.440
i look and i say well 4 to the something and 8 to the something
00:31:57.440 --> 00:31:58.360
its not gonna work
00:31:59.020 --> 00:32:02.060
cant let them equal each other but i could let 4 be 2 squared
00:32:07.140 --> 00:32:08.340
and let 8 be 2 cubed
00:32:17.840 --> 00:32:20.400
now i multiply it by the powers and i det
00:32:21.100 --> 00:32:21.900
2 to the 6th x
00:32:22.880 --> 00:32:23.380
minus 2
00:32:24.840 --> 00:32:25.480
2 to the 3x
00:32:26.140 --> 00:32:29.140
plus 3. do you see where theses numbers are coming from, youre just multiplying
00:32:30.740 --> 00:32:34.700
now i know 6x-2 has to equal 3x+3
00:32:40.580 --> 00:32:43.840
so do a little algebra
00:32:46.860 --> 00:32:47.360
x=5/3
00:32:48.700 --> 00:32:50.620
load up on your algebra folks
00:32:51.180 --> 00:32:54.620
the major reason people have trouble with calculus
00:32:54.620 --> 00:32:58.860
is not that they cant figure out whats going on but is the algebra
00:33:07.640 --> 00:33:13.800
okay have you guys do one real fast and have you move on from here to the more important content
00:33:13.800 --> 00:33:15.200
we understand this one
00:33:34.460 --> 00:33:35.100
that is a 9
00:33:50.720 --> 00:33:51.940
the 9 is 3 squared
00:33:59.300 --> 00:34:00.100
27 is 3 cubed
00:34:13.340 --> 00:34:15.900
now i know 9 is 3 squared and 27 is 3 cubed
00:34:16.760 --> 00:34:21.440
so i can multiply the powers, so i get 3 to the 2x minus 4
00:34:21.440 --> 00:34:25.160
is 3 to the 9 minus 3x
00:34:32.240 --> 00:34:33.280
that means 2x-4
00:34:35.620 --> 00:34:37.640
must equal 9-3x
00:34:39.460 --> 00:34:40.100
so i get 5x
00:34:41.000 --> 00:34:41.500
plus 13
00:34:43.400 --> 00:34:43.960
x is 13/5
00:34:44.660 --> 00:34:45.860
howd we do on that?
00:34:46.760 --> 00:34:47.560
we get 13/5?
00:34:48.680 --> 00:34:50.920
we have a feeling of satisfaction
00:34:53.340 --> 00:34:55.820
okay lets move on to something harder
00:34:55.820 --> 00:34:57.640
suppose to know them for logarithms
00:35:00.920 --> 00:35:02.880
yes such happy voices
00:35:26.160 --> 00:35:27.760
alright ready to learn about logarithms?
00:35:28.820 --> 00:35:29.800
its actually very simple
00:35:58.940 --> 00:36:00.380
alright so 10 to 1 is 10
00:36:00.380 --> 00:36:02.940
and 10 to the 2 is 100, right? we know this
00:36:04.400 --> 00:36:06.520
so it must be power of 10
00:36:08.320 --> 00:36:09.440
that gives you 50
00:36:10.880 --> 00:36:13.840
theres got to be a power of 10 that gives you 50. you just dont know what that power is
00:36:13.840 --> 00:36:16.880
but we know somewhere betwen one and 2
00:36:17.900 --> 00:36:19.740
right so here is a power of 10
00:36:20.160 --> 00:36:21.440
that will give us 50
00:36:21.660 --> 00:36:23.340
and its 1 point something
00:36:25.300 --> 00:36:26.660
so we call that power
00:36:27.840 --> 00:36:28.800
the logarithm
00:36:29.600 --> 00:36:30.100
of 50
00:36:31.760 --> 00:36:32.260
i know
00:36:34.820 --> 00:36:36.260
but heres the problem
00:36:38.320 --> 00:36:39.920
i know that 2 to the 5 is 32
00:36:41.940 --> 00:36:43.060
and 2 to the 6 is 64
00:36:43.600 --> 00:36:46.480
so there must be a power of 2 that gives you 50
00:36:50.220 --> 00:36:52.220
so we can call that the log of 50
00:36:55.620 --> 00:36:58.100
but that cant be the same log, they cant be the same number
00:36:58.100 --> 00:37:01.220
right because this is a number somewhere between 1 and 2
00:37:01.220 --> 00:37:03.600
and this is number somewhere between 5 and 6
00:37:04.660 --> 00:37:07.540
so how do we know the difference, so what we do we gonna write a number down here
00:37:09.540 --> 00:37:10.340
to remind us
00:37:10.460 --> 00:37:13.020
thats called the base of the logarithm
00:37:17.040 --> 00:37:17.760
so if i know
00:37:19.140 --> 00:37:20.020
4 to the 2 is 16
00:37:21.580 --> 00:37:22.700
and 4 to the 3 is 64
00:37:22.700 --> 00:37:25.240
then theres a power of 4 that gives me 50
00:37:26.820 --> 00:37:28.500
so thats also the log of 50
00:37:29.080 --> 00:37:30.440
so dont get confused
00:37:31.720 --> 00:37:32.680
i put a little 4
00:37:34.500 --> 00:37:36.820
so thats the based of the logarithm
00:37:38.460 --> 00:37:40.300
now i can sort of generalize
00:37:43.600 --> 00:37:44.900
lets think about whats going on
00:37:45.460 --> 00:37:49.300
what is that x? that x is some number with a decimal usually
00:37:50.420 --> 00:37:52.500
and if i take the base
00:37:53.240 --> 00:37:55.080
take the little number here
00:37:55.700 --> 00:37:57.960
and raise it to x ill get this
00:38:01.720 --> 00:38:05.780
so if i take 4, and raise it to this x ill get 50
00:38:06.020 --> 00:38:10.100
and if i take 10 and raise this x ill get 50. theyre not the same x
00:38:11.760 --> 00:38:13.120
because they cant be
00:38:13.120 --> 00:38:15.380
they have to have different values
00:38:15.380 --> 00:38:17.320
this side has to be between 5 and 6
00:38:17.320 --> 00:38:19.560
that value has to be between 1 and 2
00:38:19.560 --> 00:38:21.420
this value has to be between 2 and 3
00:38:21.420 --> 00:38:22.800
everyone understand so far>
00:38:24.020 --> 00:38:25.780
ill repeat if you need me to
00:38:25.960 --> 00:38:28.360
or you can rewind the video and watch
00:38:31.780 --> 00:38:34.040
okay so in general
00:38:40.440 --> 00:38:42.040
if i know x=log base b of a
00:38:44.620 --> 00:38:45.980
that means if i take b
00:38:46.980 --> 00:38:48.340
and i raise it to the x
00:38:49.900 --> 00:38:50.400
i get a
00:38:51.160 --> 00:38:52.120
so a logarithm
00:38:52.940 --> 00:38:55.740
and exponential are related to each other
00:38:55.740 --> 00:38:57.300
in fact theyre inverses
00:39:00.520 --> 00:39:03.000
if i put in a i get x and if i put x ill get a
00:39:08.760 --> 00:39:09.720
so far so good?
00:39:11.100 --> 00:39:13.500
so lets try an extent to a few of these
00:39:21.720 --> 00:39:23.800
i have to find the log base 5 of 25
00:39:23.800 --> 00:39:26.980
we cant ask you to find complicated numbers, you need a calculator
00:39:28.400 --> 00:39:29.760
tables or something
00:39:29.760 --> 00:39:32.920
you have no idea how to find a log, just by guessing
00:39:35.120 --> 00:39:36.720
so 5 raised to some power
00:39:38.020 --> 00:39:40.420
has to give you 25, this means i take 5
00:39:40.980 --> 00:39:42.100
and raise to the x
00:39:43.440 --> 00:39:44.080
ill get 25
00:39:46.600 --> 00:39:48.760
x must be 2 because 5 squared is 25
00:39:51.740 --> 00:39:52.540
if i had a log
00:39:55.020 --> 00:39:55.520
base 5
00:39:56.720 --> 00:39:58.440
of 125 equals x
00:39:59.340 --> 00:40:02.060
that means if i take 5 and raise it to this x
00:40:03.080 --> 00:40:03.640
i get 125
00:40:09.300 --> 00:40:10.100
so x must be 3
00:40:19.960 --> 00:40:21.220
how are we doing on thses so far?
00:40:21.220 --> 00:40:23.140
you shoulve seen these before
00:40:24.060 --> 00:40:26.260
if you knew this well enough of course you wouldnt be in this class
00:40:29.460 --> 00:40:30.820
only the night befor
00:40:32.200 --> 00:40:33.240
placement test
00:40:37.300 --> 00:40:38.420
alright how bout
00:40:48.820 --> 00:40:49.780
log base 2 of 16
00:40:50.860 --> 00:40:52.140
that says if i take 2
00:40:53.920 --> 00:40:55.200
and raise it to the x
00:40:57.040 --> 00:40:58.320
i get 16 so x must be 4
00:41:01.880 --> 00:41:03.160
get the point home?
00:41:09.900 --> 00:41:10.700
well if i say
00:41:11.960 --> 00:41:12.600
log base 3
00:41:14.140 --> 00:41:14.640
of x
00:41:19.220 --> 00:41:20.900
now im going the other way
00:41:21.880 --> 00:41:22.760
now im saying
00:41:24.680 --> 00:41:25.240
3 to the 5
00:41:33.600 --> 00:41:34.480
so that means
00:41:36.420 --> 00:41:37.880
x=243
00:41:42.800 --> 00:41:45.280
notice i can go both directions, i can
00:41:45.480 --> 00:41:47.960
know what base number is and find the x
00:41:48.540 --> 00:41:50.380
or i go look for the x and find
00:41:50.380 --> 00:41:52.500
basically the logarithm, the power
00:41:54.220 --> 00:41:55.420
do either of those
00:42:10.020 --> 00:42:12.260
so now how am i going to use the logs
00:42:20.620 --> 00:42:23.560
we know how to solve 5 x to the 25
00:42:24.700 --> 00:42:25.420
we said x=2
00:42:25.820 --> 00:42:27.580
what if i said 5 to the x is 20
00:42:28.760 --> 00:42:30.280
how would i solve that?
00:42:30.620 --> 00:42:32.620
thats what logarithms are for
00:42:32.620 --> 00:42:35.840
logarithms were invented to solve these problems
00:42:37.220 --> 00:42:41.380
because here i can just say well this is 2 because 5 squared is 25
00:42:42.220 --> 00:42:45.260
so here i have no idea what it is, its less than 2
00:42:45.340 --> 00:42:47.260
i dont know exactly what it is
00:42:47.860 --> 00:42:50.920
in fact its log base 5
00:42:51.900 --> 00:42:52.400
20
00:42:52.840 --> 00:42:54.280
so were gonna practice converting
00:43:01.340 --> 00:43:02.460
thats the answer
00:43:02.460 --> 00:43:04.640
i dont care what the actual number is
00:43:06.120 --> 00:43:07.160
thats one point
00:43:07.860 --> 00:43:08.840
88 or something
00:43:10.380 --> 00:43:11.020
who knows
00:43:12.420 --> 00:43:13.900
whip out your calculator and find out
00:43:16.360 --> 00:43:19.400
when we give you these problems, this would be a perfectly good answer
00:43:20.080 --> 00:43:21.840
we wont expect you to put it
00:43:26.700 --> 00:43:27.980
so if i have 7 to the x
00:43:28.940 --> 00:43:29.440
is 100
00:43:30.560 --> 00:43:32.820
x would be log base 7, 100
00:43:34.900 --> 00:43:39.180
you get and idea how big it is because 7 squared is 49 so its got to be bigger than 2
00:43:39.180 --> 00:43:44.200
7 cubed is less than 343 so its got to be less than 3, so it has to be somewhere between 2 and 3
00:43:47.140 --> 00:43:48.020
so far so good
00:43:53.820 --> 00:43:58.780
this is how you can use a log to solve various simple exponential equations
00:43:59.180 --> 00:44:02.300
so lets figure out some stuff we can do with logs
00:44:17.400 --> 00:44:19.160
so what is the log of 1/2 of b
00:44:23.960 --> 00:44:25.000
how do you know?
00:44:28.800 --> 00:44:31.840
log base b of 1 is 0 because anything to the 0 is 1
00:44:31.840 --> 00:44:32.540
wed advanced
00:44:35.220 --> 00:44:40.980
if i take b and raise it to the 0 ill get 1, remember thats what the equation tells us rught
00:44:41.700 --> 00:44:43.380
b to the 0 is 1 well
00:44:43.440 --> 00:44:44.800
anything to the 0 is 1
00:44:45.460 --> 00:44:47.540
so log of 1 is always going to be 0
00:44:47.920 --> 00:44:50.160
remember this for calculus okay?
00:44:51.660 --> 00:44:55.020
log of 1 is 0 doesnt matter what the base is
00:45:00.600 --> 00:45:02.040
whats the log base of b
00:45:07.000 --> 00:45:07.500
1
00:45:08.680 --> 00:45:11.460
why is it 1? well b to the 1
00:45:13.200 --> 00:45:13.700
b
00:45:19.420 --> 00:45:20.620
how bout log base b
00:45:23.320 --> 00:45:24.120
b raised to x
00:45:26.240 --> 00:45:27.680
thats gonna come out x
00:45:31.100 --> 00:45:32.240
because thats really just saying
00:45:32.560 --> 00:45:34.800
i take b, what power do i raise it to
00:45:35.880 --> 00:45:37.960
to get b to the x i raise it to the x
00:45:38.440 --> 00:45:41.000
this is how you know these are inverses
00:45:44.080 --> 00:45:46.780
that equation helps you figure out what there inverse
00:45:47.500 --> 00:45:49.820
okay you also need to know the graph
00:45:49.820 --> 00:45:51.880
not gonna worry about it exactly
00:45:54.320 --> 00:45:56.220
when i raise something to the power
00:46:00.080 --> 00:46:02.440
and i take a base and raise it to a different power
00:46:03.260 --> 00:46:05.260
i add the powers remember that
00:46:07.660 --> 00:46:09.580
so the logarithms are powers
00:46:11.160 --> 00:46:12.280
so if i had the log
00:46:13.720 --> 00:46:14.220
of ab
00:46:15.960 --> 00:46:17.400
we multiply together
00:46:18.460 --> 00:46:19.980
and we just add the logs
00:46:24.100 --> 00:46:25.780
so thats a very important
00:46:26.200 --> 00:46:28.040
rule, thats called a log law
00:46:28.080 --> 00:46:29.680
and thats law number one
00:46:32.540 --> 00:46:33.580
theres 3 of them
00:46:39.480 --> 00:46:41.080
if i draw a box around it its probably important
00:46:43.180 --> 00:46:43.820
maybe not
00:46:44.920 --> 00:46:46.280
what if i had a log of a
00:46:48.280 --> 00:46:48.780
over b
00:46:51.840 --> 00:46:53.840
well thats gonna be the log of a
00:46:54.900 --> 00:46:56.020
minus the log of b
00:47:13.860 --> 00:47:17.860
just like when you multiply you add the logs, and when you divide you subtract the logs
00:47:18.420 --> 00:47:20.020
even if i have the log of a
00:47:21.200 --> 00:47:22.160
raised to the b
00:47:23.700 --> 00:47:25.140
thats gonna be b times
00:47:27.560 --> 00:47:28.060
log of a
00:47:29.800 --> 00:47:30.360
because
00:47:31.020 --> 00:47:32.100
im gonna multiply it by b
00:47:33.680 --> 00:47:36.620
you guys should have seen this back when you first started logs
00:47:37.320 --> 00:47:38.600
ill raise it up then
00:47:43.420 --> 00:47:45.580
so these are very useful to learn
00:47:47.800 --> 00:47:49.320
especially that last 1
00:47:55.140 --> 00:47:55.700
suppose
00:48:03.100 --> 00:48:04.700
i have 3 to the x equals 40
00:48:05.640 --> 00:48:07.400
now i know that says the log
00:48:07.840 --> 00:48:09.360
that x is log base 3 of 40
00:48:09.360 --> 00:48:11.520
whats another way you can do this?
00:48:11.520 --> 00:48:13.460
you know how you can square both sides
00:48:13.460 --> 00:48:17.400
and square root both sides and things of both sides, you can take the log of both sides
00:48:22.840 --> 00:48:25.320
you should see logs a lot in chemistry
00:48:27.140 --> 00:48:28.580
log scales, acid base
00:48:30.740 --> 00:48:33.220
what are you guys doing in chemisrty?
00:48:55.380 --> 00:48:57.140
i take the log of both sides
00:48:57.940 --> 00:49:01.140
but i put the x in front because the log of a to the b
00:49:01.900 --> 00:49:03.020
is b times log of a
00:49:03.640 --> 00:49:04.760
so log of 3 to the x
00:49:05.600 --> 00:49:06.320
is x times'
00:49:07.980 --> 00:49:08.480
log of 3
00:49:11.300 --> 00:49:14.340
and remember log of 3 is just a number
00:49:14.340 --> 00:49:16.520
right so some number with a decimal
00:49:16.520 --> 00:49:17.980
and thats just some number
00:49:18.460 --> 00:49:21.440
so x is log 40
00:49:23.480 --> 00:49:24.120
over log 3
00:49:24.120 --> 00:49:25.780
so if this were a test question
00:49:25.780 --> 00:49:27.980
that would be what we were looking for
00:49:29.940 --> 00:49:32.020
if you say 3 to the x is 40, whats x
00:49:32.860 --> 00:49:34.220
log of 40 over log of 3
00:49:35.400 --> 00:49:35.900
or
00:49:36.000 --> 00:49:38.000
you could of said like base 3 40
00:49:38.000 --> 00:49:39.280
theyre equivalent
00:49:43.320 --> 00:49:44.760
so again lets say i had
00:49:48.900 --> 00:49:49.780
7 to the x is 20
00:49:50.540 --> 00:49:52.220
take the log of both sides
00:50:02.900 --> 00:50:05.620
i can multiply the x in front and get x log 7
00:50:07.420 --> 00:50:08.540
with the log of 20
00:50:10.360 --> 00:50:11.160
if i divide
00:50:13.740 --> 00:50:14.240
log 20
00:50:15.800 --> 00:50:16.680
by the log of 7
00:50:47.780 --> 00:50:48.980
if i take the power
00:50:48.980 --> 00:50:51.780
and multiply the front multiply the bottom
00:50:51.960 --> 00:50:55.240
i take the power put it in front and multiply by log
00:50:57.500 --> 00:50:58.140
thats all
00:51:00.480 --> 00:51:03.440
not the same as some of these other equations
00:51:06.920 --> 00:51:09.800
so lets move on to one slightly more messier
00:51:22.860 --> 00:51:23.980
what if i had that
00:51:35.560 --> 00:51:37.560
i can take the log of both sides
00:51:45.220 --> 00:51:46.820
and i put the x+3 in fornt
00:51:47.540 --> 00:51:49.460
dont forget the parenthesis
00:52:02.160 --> 00:52:04.320
okay now lets figure out what x is
00:52:05.800 --> 00:52:07.240
so i can divide by log 6
00:52:07.840 --> 00:52:08.800
and subtract 3
00:52:11.420 --> 00:52:12.140
and get x+3
00:52:14.000 --> 00:52:14.560
log of 35
00:52:16.120 --> 00:52:16.920
over log of 6
00:52:20.500 --> 00:52:21.460
so x is log of 35
00:52:23.680 --> 00:52:24.480
over log of 6
00:52:26.740 --> 00:52:27.240
minus 3
00:52:29.600 --> 00:52:31.040
you can say its tricky
00:52:31.900 --> 00:52:35.240
you have to learn the simple work before you learn the hard ones, you understand that?
00:52:48.680 --> 00:52:52.600
the log of 55 over the log of 6 is not the same as the log of 35/6
00:52:54.180 --> 00:52:57.060
becareful doing that on a test or webassign
00:52:58.760 --> 00:52:59.720
the log of 35/6
00:53:01.660 --> 00:53:02.780
is the log of 35-6
00:53:03.720 --> 00:53:05.240
dont confuse these two
00:53:05.980 --> 00:53:09.820
theres another rule that im gonna show you guys next times
00:53:11.280 --> 00:53:12.000
which will
00:53:12.000 --> 00:53:14.980
to show you what to do with one logarithm and the other
00:53:15.080 --> 00:53:16.840
so see everybody on monday