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Logarithms & Logarithmic Functions
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May 17, 2025
In this interactive Algebra 2 lesson, we dive deep into the world of logarithmic functions—breaking down everything from the parent function to transformations, domain/range, and graphing techniques that make logs easy to understand. What you’ll learn in this video: ✅ What a logarithmic function is and how it's connected to exponents ✅ How to graph basic log functions like f(x) = log₅(x) and log₁₀(x) ✅ How to find the domain, range, and vertical asymptotes ✅ How to use a calculator to evaluate logarithms using the change of base formula ✅ How transformations (like 3log₁₀(x) + 1) affect the graph ✅ Why the graph behaves differently when the base is less than 1 ✅ The step-by-step process of building a table of values and sketching accurate graphs ✅ Real classroom questions and student interactions to reinforce learning
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0:00
yes so we talk about uh now it's a word
0:04
that I have a hard time saying so I'm
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not I'm just going to call it log all
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right i'm not going to say it's supposed
0:10
to be log but I'm just going to say log
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when I say log understand that this what
0:15
I'm referring to all right so we're
0:17
going to talk about the log function and
0:19
log properties thank you so it's going
0:21
to take us 7.3 4 5 6 to finish this so
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each one of them is going to be
0:26
different so log applications are pretty
0:30
much exponents this is another way of
0:33
writing exponents right so you're going
0:35
to see it's called a lo don't mess I'm
0:38
turning off so it's the logarithm
0:40
function right so they are pretty much
0:42
exponents so example if I have
0:47
b^ of x = to a this means that log base
0:54
b a is equal to x so here when I put the
0:58
uh the letter B means the base all right
1:01
log base A is equal to X that's what
1:04
that means right so B to the X = A means
1:08
that log base B A is equal to X so
1:12
example if I use actual numbers if I put
1:15
2 to the 3 is equal to 8 this means that
1:18
the log base 2 of 8 is 3 that's what
1:21
that means this is how you're going to
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read this log base 2 of 8 is three or if
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you want to rewrite it you can just say
1:27
23 is equal to 8 so log is pretty much
1:31
another way of writing exponents all
1:33
right so this is what this this means so
1:36
b to the x = to a means that log base b
1:40
right the number that is under the g is
1:42
called a base oh all right okay so if I
1:47
say 23 equals to 8 if I say write this
1:50
using log you're going to say log base 2
1:53
of 8 is
1:55
three so this is how we use a log
1:58
function you have it in your calculator
2:00
right so the first thing that we're
2:02
going to learn how to do is we're going
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to have we're going to learn how to go
2:07
from the log uh base to the exponential
2:10
form all right so we can switch back and
2:13
forth example the question is is this
2:16
write each uh equation in exponential
2:20
form so write each log equation in
2:24
exponential form so I have log base 2 x=
2:27
5 so how would you write this using just
2:30
exponent so this is why I put the area
2:33
here so how would you write this using
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the same method that I use here
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uh is
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it okay mhm just a guess
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is it 25 no 2 to the Wait fifth 2 to the
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5th equals what x x that's it
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2 to the 5th b to the X= A means log
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base B is equal to X i thought you
2:58
trying to find X right x is 2 to the 5th
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oh
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I'm just rewriting this using exponent
3:07
so these are equivalent it's the same
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thing so log the log function is also a
3:12
way of writing exponential right so log
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base 2 of x is 5 means that 6 is equal
3:18
to x
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okay okay so it's kind of abstract and
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sometimes you don't see it so basically
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this is the base right this is the
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exponent and this is the number that it
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gives you so that means two if log base
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2 of x is 5 that means 2 ^ 5 is equal to
3:35
x all right so you just have to
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recognize that this is the base this is
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the exponent and this is the number that
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it yields to all right yeah um the base
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like won't ever change right well the
3:50
base won't ever change right so here how
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to write this using the same thing
3:59
to the what
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equals 256 not 256 1 256 right so that's
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what it means this means 4 ^ of -4 = 1
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256
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right this is the base right so let's
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break it down this is called the
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base this is the
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exponent and this is the
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number that it yields to right he yields
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to or he gives so that's what that means
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okay you have your base your exponent
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and what you get when you raise it to
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the power of that exponent right so how
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would you write this math uh let me put
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this uh here so how would you write this
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x^ 2 is equal to what x where's X i mean
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16 goes under that it ain't here how
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would you do it
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yeah
5:01
= 79 729 so that's basically what a log
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function is it's an exponent okay you're
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just using this to write an exponent
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right so now we're going to go the other
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way around now we're going to go for the
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exponential form to the log form right
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so half is 15 to the^ 3 is equal to
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3375 so it be log base what log is base
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is the little number on the bottom or
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the top the base it would be Yeah now
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this is the base right okay the base so
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the base is going is going to be what
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15 right and then the number is going to
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be 3 375 and the exponent is going to be
5:41
three that's all it's just learning how
5:44
to rewrite this right so now here 15 to
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30 = to 3375 means log base 15 of 3375
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is three okay I can do the next one okay
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now what's the next one okay log base
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for
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um 12 or equals two oh no yeah okay and
6:07
then one half right
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and then two right okay what about this
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one here me again mhm okay um
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log base 4
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um = 3 and then 64 is the number all
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right that's it right cool
6:34
log base what's the base
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what's the base 125 right equals what
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why do I need to do it in that order
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well what I didn't know where the
6:46
numbers were going you were saying
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numbers i don't know no look log five
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okay you mean log But you have to tell
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me what the base is first yeah why so I
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know cuz I when you say log five I don't
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know where you want me to put the five
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all right all right now we're going to
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learn how to evaluate the log expression
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right after evaluate the log expression
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now here's what I have here i want to
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solve for x right so what I have is log
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base 16 to the^ 4 and I want to evaluate
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this i want to find what this is right
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i'm trying to solve for x i don't know
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what x is so here's what you do you've
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been given the log of 16 base 16 of 4 we
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don't know what it is so we're going to
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call the number what x i don't know what
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it is but I have to solve for it correct
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i need to solve it so I don't know what
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it is i need to solve it so I have log
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base 164 is equal to X this is where I
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start i want to evaluate this i want to
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find what it gives me evaluate is find
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what it gives me right so now we know
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that we can use a transformation here
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right it's going to be what 16 to the^
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of what
7:50
equals to four now do we know how to
7:52
solve this yes yes we did it in the last
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chapter right so what are to this one
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here
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good question excuse me i want to break
8:02
it down 16 what the base is what
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four so it's going to be four to the
8:07
power what two to the power of what x =
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four 4^ 2 1 right so this is 2x
8:19
= 1 and then x = what 12 so x is 12 okay
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right
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okay so we start with a vague expression
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log base 416 we don't know what it is
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but we can find it so we're going to
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call the number that we are looking for
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x because we don't know what it is but I
8:39
know that based on the change of base
8:40
formula which we just learned we know
8:42
that log base 16= to x means that 16 to
8:47
the^ x is equal to
8:49
4 now in the previous chapter we learned
8:52
how to solve exponential functions so I
8:54
can turn 16 into 4 ^ 2 so that gives me
8:58
4 ^ 2x = 4 ^ 1 4 is 4 the^ of 1 now
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because the bases are the same I can
9:05
drop the base and just solve the regular
9:07
problem 2x = 1 so x is 12 right so I go
9:11
back here log base 16 of 4 is basically
9:14
what 12 right so let's do another
9:18
problem
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here so how would you evaluate uh log
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base 3 can I erase this
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okay so how do we get
9:32
log base 3 of 81 so let's find and also
9:38
find log base
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um log base
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12 of 256 so how we going to do this so
9:49
take about like two minutes trying to
9:51
solve it
9:54
so we have log base 3 of 81 and log base
9:58
1/2 of 236 let's try to solve it
10:10
mr
10:22
yeah that's good yeah and do the next
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one
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what did we get for the first one
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I got x= 4 so tell me how you did it so
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first
10:59
= 81 all right and then broke down the
11:01
81 so it's 3x = 3 to 4 okay and then x =
11:07
4 x = 4 all right good then you just do
11:10
like 3^ 1 2
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3 that's that's that's um sort of like a
11:18
little tire that's what if the
11:20
expression is bigger than that you're
11:22
going to spend a whole day and still
11:23
can't figure it out we have to do it
11:25
like uh uh we have to use the method
11:27
that we've learned how how about this
11:30
one so what did y'all do here uh I give
11:32
up mhm i don't know how to break so what
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do we do so let's start what
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uhhuh x to X to the X equals 256 256 all
11:44
right so now I have 1/2 here right and I
11:48
have 256 why do you think the base the
11:50
accurate base going to be here base what
11:55
what what base
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what do you mean what base you don't
11:59
know what the base mean figured it out
12:01
yeah what base did you use figured it
12:03
out this is one half so what do we do
12:05
the base is two two two to the power of
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what this is
12:10
12x thank you two to the^ ofx very
12:14
simple stuff we did this 12 just change
12:17
to a negative exponent and a 26 is what
12:20
2 to the^ of what 2 the^ 8 2 to the^ of
12:23
8 and now we can solve it right sox= 8
12:26
that means x is equal to what 88 pretty
12:30
simple stuff right just got to like
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think a little bit sometimes we get hung
12:35
up on stuff because you're not thinking
12:37
all right so I have log base
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6 right log base
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6 of 1 =
12:47
to I want to evaluate this so how do I
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do this log base yeah
13:00
6 to the what
13:06
6 or 6x equals what all right that's not
13:11
happy
13:14
mhm
13:17
question think about it what base are we
13:20
looking for base of what
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b of one this is one so we have to
13:26
multiply them both this is base of what
13:28
what's base here so therefore this has
13:30
to be base six so how can I turn this
13:32
into a base six
13:35
no six to the power of what
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no any number of what is one what we
13:44
talked about it
13:48
zero thank you
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right so therefore x is what zero come
13:56
on
13:58
come on
14:04
all right all right so with that said
14:07
we're going to close off section one cuz
14:09
section tomorrow we're going to talk
14:10
about solving uh I'm actually graphing
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log functions now we just s talking
14:17
about evaluating and expressing them so
14:19
go ahead and then start now okay that's
14:22
what we're going to talk about so we're
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going to learn how to graph log
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functions yesterday's uh section was
14:28
super easy i love the fact that you guys
14:30
got your stuff done in an exponential
14:33
speed and I'm happy so now we're going
14:34
to continue we're going to learn how to
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graph like log functions okay they are
14:38
not hard at all they're just like the
14:40
exponential functions and like I said
14:42
yesterday logs are pretty much exponents
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right so now how do you graph a log
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function so the first thing you want to
14:49
know is we want to know what the parent
14:51
function is right parent function is the
14:53
function that starts it all and then out
14:55
of that we can graph any other function
14:58
right so the parent function looks like
15:00
this log b x right log base x okay and
15:06
the base is never equal to one that's
15:09
why I put this here b not equal to one
15:12
means the base can never be equal to one
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right so log now in our in our everyday
15:18
language do you know what log base we
15:20
use base what in all the numbers that we
15:23
use x no base what
15:27
zero base 10 10 why that's That's what
15:30
we use the base 10 is what we use on a
15:32
regular basis log base 10 right so we
15:34
use that so now wait for example one
15:38
right one can be written as what one is
15:41
basically log base 10 10 that's what it
15:44
is we're going to get to that in the
15:46
next chapter but I just want to tell you
15:48
we use the base 10 so now example I'll
15:52
give you an example here we want to
15:53
graph this function right the function
15:55
is f ofx is equal to log base 5 of
15:58
x we want to graph practice function so
16:01
what do we do we have some steps that we
16:03
have to take there are specific steps
16:05
that I would like you to follow and if
16:07
you follow these steps every function of
16:09
yours is going to be done easily right
16:12
the first thing is this the domain of
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any base b function with a x is always
16:20
going to be x is greater than zero
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because logs cannot be negative right
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you can't have log of a negative number
16:27
it does not exist you can't have log
16:30
base of 1 that does not exist so all log
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functions have to have a positive domain
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right so in this case if log x so you
16:40
just want to make sure what whatever
16:42
expression is here is greater than zero
16:44
now we're going to have some functions
16:45
that are going to have different things
16:46
like x + one you're going to have you
16:48
you should be able to like figure out
16:50
that to find the domain you want x + one
16:53
to be more than zero but here we're
16:55
going to start with the basics right so
16:57
you want x to be greater than zero and
16:59
the range of
17:00
most any log function is always for
17:04
negative infinity to positive infinity
17:06
that is another one that you need to
17:08
know right now what's your very first
17:10
step to graph this function so we need
17:12
to identify the base so what's the base
17:15
of this function here
17:18
five base five so now we identify the
17:20
base the next step is to make a table of
17:22
values and it's the easiest thing to do
17:25
right to find the table of value i'll
17:27
give you the this the basics you just
17:29
need to when you build your table right
17:33
if you want to build your table what you
17:35
want to do is this you want to always
17:37
use these three values here to avoid any
17:40
problem you always want to use one over
17:42
B B and then one okay so those three
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points are always going to help you to
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find your function so here since B is
17:53
five I start with what 1 over 5 and then
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five because that's my base and then one
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right so if you put in your calculator
18:01
log base 5 of 15 because you have to
18:04
plug it in back in here to find the
18:05
value because we are technically finding
18:08
f of one over five that's what we're
18:10
finding we're doing this log 5 of 1 over
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5 is going to give you negative 1 right
18:15
it's going to give you negative 1 and if
18:18
you put in your calculator log five of
18:20
five is going to give you positive one
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if you plug in your calculator TI3 I'm
18:26
talking about right and then if you put
18:28
log five of one it gives you zero now
18:31
has anybody has trouble putting in your
18:33
calculator
18:36
all right so here's what you can do let
18:38
me show you how to do it
18:42
let me show you how to do it right so
18:44
I'm going to show you how to do this
18:46
hold on let me I'll give it to you
18:47
because I don't want you on the
18:51
camera all right so to find log base 5
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so here's what you do so if you have a
18:59
t3 right to put this in your calculator
19:01
log base 5 of 1 over 5 here's how you do
19:05
it you're going to go log of 1 5 over
19:09
log of five this is how the calculator
19:11
reads this okay so you see the log key
19:15
on your calculator you see so put log 1
19:19
over 5 / log 5 and you should get 1 i
19:24
did you got it right all right good now
19:26
do the next one wait I got.53
19:31
i don't know why you did that did you
19:32
put it as this huh log one over five
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you put in parentheses
19:38
don't use your your That's not yet what
19:41
okay let's try this so log one over five
19:44
put in parentheses over log of five i
19:47
got 488 let me see how you put it in
19:54
i see dividing you multiply you have to
19:58
actually you have to plug plug it in
20:00
like this the way I put it log one over
20:02
five is log one i got an error for the
20:07
last one log of one
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so did you do log one okay hold on a
20:12
second divide by log of one yeah no log
20:14
of I have five divided by log of five
20:17
that should give you zero
20:20
i got zero you got zero right yeah I got
20:22
zero all right so everybody should get
20:24
the same thing right if you put you have
20:27
to make sure the base is always under
20:30
right if I have log base 5 of 1 over 5
20:33
it's going to be log of 15 over log of
20:36
five and that should give you the
20:38
negative one all right everybody should
20:40
get the same thing did you get that all
20:42
right good so now that you have your
20:44
values the rest is just formality we
20:47
just going to fill the table so when x
20:49
is so I'm going to put my values here
20:50
right so when x is 1 over 5 1 over 5 is
20:54
somewhere really close to this right 1
20:55
over 5 when x is one over 5 y is
20:58
negative 1 so right here right when x is
21:01
five 1 2 3 4 5 y is one so right here
21:06
okay and when x is
21:09
one y is zero so it's right here so now
21:12
to graph your function we say that the
21:15
the range is from what negative infinity
21:17
to positive infinity and the domain is
21:19
from zero to positive infinity so your
21:21
function is going to look like this okay
21:24
so it's going to look like this with
21:25
your three point this is called an
21:26
asmtote i talked about it last time
21:28
right you're never going to cross this
21:30
line why why can we never cross this
21:32
line
21:33
here somebody thinks about it tell me
21:35
yet is it because the law can't be
21:37
negative that's right because the law
21:39
can't be negative so therefore we can
21:41
never cross this line so it has to be
21:43
close to this line here we can never
21:46
cross this line it has to be this line
21:48
is going to serve as your border right
21:50
your border or your asmtote okay so this
21:52
is how the function is going to
21:54
look all right so now let me ask you
21:57
guys a question if I was to graph log
22:00
base uh 3 of x if I want to graph this
22:06
function how would you make your table
22:08
somebody tell
22:09
me what would your table look like
22:14
133 okay three and one one that's it so
22:19
with this function that's all I want you
22:21
to do all right now let me ask you a
22:23
question i want to see if you guys are
22:24
following me here so what if I have log
22:27
base
22:29
1 of x so how would you make your table
22:32
now five five and then
22:36
1/5 and one one it's the same thing
22:39
we're just reversing the I mean you
22:40
could just start with 15 too it would be
22:42
the same thing right we will get the
22:43
same result okay so now with this
22:46
function let's try to graph this let's
22:48
try to graph this function here with the
22:49
base of 1 over 5 so when x is five how
22:53
you going to put in your calculator it's
22:55
going to be log five over
22:57
what log of 1 over 5 and tell me what
23:01
you
23:03
negative one good and then this is going
23:06
to give you automatically
23:09
what that will give you
23:12
positive one right and then the other
23:15
one is going to give you what you
23:17
already know
23:19
zero zero thank you and now let's graph
23:22
it right see there's going to be
23:23
something different here when x is five
23:25
1 2 3 4 5 y is negative 1 right here
23:30
right and then when x is
23:33
uh 1/5 y is one so right
23:37
here and then when x is one y is zero
23:41
right so now look at the function it's
23:43
going to look like
23:47
this right you see you have the same
23:50
problem not the same problem we have the
23:52
same result but it's just reversed right
23:55
again we're not going to get past this
23:57
line why just like what Patrick say you
24:00
can't go over the negatives right but
24:03
the difference is this when the base is
24:05
less than one the graph goes down this
24:08
way when the base is more than one it
24:11
goes where it goes the other way does
24:13
that make sense right we just need to
24:16
graph a couple of those and it's going
24:17
to get easier all right so now we're
24:20
going to learn how to graph uh this log
24:22
function when there's added uh
24:25
information for example let me say I
24:28
wanted to draw
24:39
so let's say we have this problem here
24:41
we want to graph this function f ofx is
24:44
three log base 10 x + one right let's
24:50
say this is what we have the book gives
24:51
us this so how do you how would you
24:53
graph this so what what do you see
24:55
that's different now three in front of
24:57
the wall the three and then what else
24:58
plus one plus one so what comes into
25:02
your head like what do you think we
25:03
should do right now to make this easy
25:05
yeah
25:07
uh okay mhm well I don't know i would
25:11
say do just the log by itself thank you
25:13
so just divide the three for the whole
25:15
equation first we start with the log by
25:17
itself right we're going to start with a
25:18
log by itself we're going to isolate
25:19
this guy we're going to start with the
25:22
log by itself okay so we're going to
25:24
start with this and make a table of
25:25
values here
25:27
right we're going to start with this so
25:29
what values am I going to use here
25:33
10 one over 10 10 one thank you so
25:37
what's this going to give us you already
25:39
know
25:41
one what's it going to give us one zero
25:45
right are they always It's probably
25:47
going to be like that for the most part
25:49
right because it's log base 10 x every
25:51
time we have a x that's going to happen
25:53
now the question that I have for you is
25:54
this we all we have is this but what do
25:58
we want three times right three times
26:01
that we just have what we have here is
26:03
just log base 10 x right but I want you
26:06
how many times three times so what value
26:09
am I going to multiply by three x or Y's
26:11
x y y's right no wives wise
26:17
wise
26:19
i said
26:22
so we have one over 10 right 10 and one
26:26
so three times that right so that means
26:28
I'm going to change the uh these to what
26:31
what's that going to be here three and
26:35
zero so now I just need to graph this
26:37
right what about the plus one oh we see
26:39
you see we go to the plus one
26:42
move it up one unit right
26:45
so 1 / 10 and -3 so 1 2 3 somewhere here
26:49
right and then 10 1 2 3 4 5 6 7 8 9 10
26:54
10 and three 1 2
26:57
3 and then one and zero somewhere here
27:00
right so we have this guy and now I'm
27:03
going to move it up how many unit one
27:05
one so this guy's going to go up one up
27:08
one and up one and then we got a new one
27:12
and that's it so again
27:15
What three this
27:17
are you hearing me i was trying to put
27:20
my calculator oh well we started with
27:22
first we graphed this and then we
27:23
multiplied the y by three and then we
27:26
added the one by shifting shifting it up
27:29
all right does that make sense are we
27:31
good on that all right so this is
27:33
basically this this section like I told
27:34
you it's not going to be that hard so
27:36
I'm going to give you some work again
27:37
and we're going to we're going to
27:38
terminate this all right so let me give
27:41
you some problems in the book and then
27:42
we're going to actually do it