>> Alright, so let's talk
about logarithms just briefly.
Okay? I'd like you to
have an introduction
so you can just start
the homework, at least.
Okay? So, maybe the
first place to start here is
that there's a little bit
of symbology here that kind
of is easy to bogged into.
So I want to phrase this in
kind of a slightly different way
than your book and most
people try and phrase it.
Okay? When you see a
logarithm, I want you to think
that this is a function,
right, first.
And specifically
an inverse.
Okay? If you think these things,
hopefully we can avoid some
of the errors that are pretty
common with logarithms.
Okay? The idea is,
right, from section 11.1,
I had these functions that we
called exponential functions.
And maybe we're just
kind of recalling this.
So exponential functions
were these guys.
They look like f of x was
some number to the x. Right?
We talk about these
for half lives
and bacteria growth
and other stuff.
Right? Compound interest
and whatnot.
Okay? So that's these guys.
I would like a way
to undo those guys
because they're used
so commonly.
Right? Basically I would like a
way to solve all the equations
from 11.1 without
bashing my head
against the wall
about changing bases.
Right? Okay, so let's think
about an undoing
function for these guys.
Cool? So my goal is invert
exponential functions.
Okay? To do that, we
just make up a rule
that says undo exponential
functions.
Okay? So here's my rule.
I'm going to say y
is log base b of x,
and I know this is
scary, right, okay?
>> Is that p or that --
>> This is a b. There's
a what for this?
>> Isn't there a log
key on a calculator?
>> Yeah, there's a key for
this on your calculator.
So let's think about
the pieces here.
So I want you to say this.
Right? So we need
to way to say this.
So we say y is the
logarithm, maybe.
Cool? I'll abbreviate that
word "log" fairly often.
Most books will write
out logarithm.
Cool?
>> What's the b, log b of x?
>> So y is the logarithm base
b. Okay, so there's going
to be a reason for
the base b of x. Okay,
so this is how we say this.
Cool? And then what we
think is we look at this
and we think this whole
business here, right, the log
and the b part, that's
the name of a rule.
Okay? So this is
a function's name.
Right? So think about this as an
elaborate way to write f. Cool?
[ Inaudible Student Question ]
Yeah, we're getting there.
It's the undoing
of the exponential function
but we'll get there.
But right now I just want you
to think this thing
is an elaborate way
to say f of x. Right?
So don't cancel the x's.
Cool? You guys good with that?
Some of you are going
to cancel the x's.
I will yell like mad.
That thing is a function.
Right?
>> Are you going to
throw stuff at us?
>> I won't throw stuff at you.
No, I probably won't because
you'll be taking a test
and I'll only be
angry at one period.
So I'll throw stuff at
the walls or something.
Cool?
>> Hey, come look at this?
>> So all right, this
is a function name.
Right? Okay, so I need
a rule for my function.
So the rule for log
base b of x, okay,
in other words how this thing
does its magic, is we say, okay,
y is the answer if
and only if, right,
so y is the answer
if b to the y is x.
Okay? So you can think about
this thing as asking a question.
Right? The question this thing
asks is what power do I raise b
to, to get x.
You guys see that?
Alternatively, you
could ask something like
what power of b is
x. So whichever one
of those resonates
more with you.
[ Inaudible Student Question ]
I usually think the second one,
but you guys can kind
of think whichever.
It's okay.
For some reason,
that phrasing seems
to be a little bit
unique and that's okay.
>> So basically just
changing the names of steps,
is that what you're saying?
>> So this is undoing, right?
So when I had f of x was a to
the x, when I plug in an x,
I just multiply that
many a's together.
Right? Here, I'm asking a
question that's a little bit
like the square root.
Right? It's not quite exactly.
But it's asking a
similar question.
Right? It's an undoing.
What did you have to do, right,
what power did you
raise b to, to get x?
>> Can you show us an
actual example of that?
>> Yeah. Let's do
an example or two.
Otherwise, I'm just
saying a bunch of words.
Okay, start easy.
So what power of 2 is 4?
Two, right?
Everybody's on board with that?
Okay --
>> So basically just [inaudible]
square root of powers?
>> So what power of 10
is the square root of 10?
[ Multiple Speakers ]
Say that again.
>> One half.
>> Why, why is it a half?
>> Oh, because that's
the square root.
>> Yeah, because this
square root is a half power.
Right?
>> So is this a lot like 11.1?
>> This is a lot
like solving 11.1.
Right? Really, this is
asking about undoing,
11.1 a lot of those questions
where they were asking solve
this little equation, right,
they were really asking you
about undoing an
exponential function.
Okay, so --
>> Is the problem
finished there is there --
>> Yeah, that's the answer.
Here, we're evaluating
a function.
Right? So this thing
is asking root 10 is 10
to the question mark.
Right? That's an 11.1 problem.
Right? What's the power?
Well, square root of 10
is 10 to the 1/2 power.
>> Because that's
the rule, right?
>> Yeah, that's the rule.
You guys see this?
Okay, should we try
some harder ones?
Just checking.
Okay? I want to make sure
they're of the right order.
I have a tendency to just
log things in any old order.
[ Inaudible Student Comment ]
No, we wouldn't want to
log things in any order.
They're a function, right.
Order matters.
What? One.
Why is it 1?
What power do you
put on 27 to get 27?
One.
>> Oh, it's right there.
>> How about log base 5 of 1?
>> Five.
>> Zero?
>> Why is it zero?
>> Because the zero --
>> Because the answer --
Right, I'm asking what power
did I raise 5 to, to get 1.
>> Five to the zero.
>> Five to the zero is 1.
Right? Are you guys
cool with this?
Do you want to see
a super weird one?
>> Zero?
>> Undefined.
>> Why is it undefined?
[ Inaudible Student Comment ]
Yeah, you can't raise 2
to any power to get 0.
Right?
[ Inaudible Student Comment ]
Another way for me to
think about this, right,
is as an inverse function.
Right? So I could graph
2 to the x. Right?
It looks like this.
If I flip this over
the line y equals x.
>> I still don't know how --
>> So this is f of x is
2 to the x. And I want
to take this picture,
right, and flip it that way.
So I get this picture.
>> Okay, would it be a
[inaudible] logarithm?
Are there actual
numbers that go with that
or it's just like an f?
>> It's a function.
Yeah, it's like, it's a rule.
>> Okay, I was thinking
that there were numbers
that [inaudible].
>> No. It's a function.
It's actually a family
of functions, right,
because the b can change.
Right? But really the
logarithm is a function.
It's that inverse
function of b to the x.
>> How do we get that
in our calculator?
>> Well, we'll get to the
calculator usages in 11.4;
11.4 is actually play
with your calculator,
figure out how logs work there.
>> So we should bring
our calculator?
>> Yeah, so it would be
good if you had a calculator
for the Monday class
because it's Friday today.
Monday is next.
Cool? This is just an intro.
Any questions on just the intro?
This should prep you to
try about the first half
of the logarithm section.
So try that homework
for the first half.
