Evaluate and Graph Logarithms (Part 1)

[INSTRUCTOR] All
right, logarithms.

Logarithms, why we need
them is exactly this right

here, what we're gonna see.

If we're trying to solve for x

And we have 2 to some
power is 16, I can think,

okay, well, 2 times what?

2 times itself how many times?

Well, 2 times itself
four times is 16,

and so x has gotta
be equal to 4.

That one's not bad when
it works out equally.

However, 2 to what power is 20?

Well, we just said 2
to the fourth was 16,

and we said we know that 2
to the fifth then is times

another 2, which is 32.

And so 20 falls someplace
in between there.

And so we know that x is
someplace in between 4 and 5.

But how exact can we get
without saying, okay, well,

2 to the 4.2, 2 to the 4.3,
4.4, and doing that,

we need logarithms.

Same deal with this one.

We know that 2 to the fifth
power, like we just said,

was 32, and we know that 2
to the sixth power is times

another 2, which is 64.

And so we know that 2 to
the 5 point something,

that x is in between
5 and 6 for this one.

But what exactly we
need logarithms for?

So, let's explore the logarithm
button a little bit here.

So 10 to the 0,
anything to the 0 power

is 1. 10 to the first,
10 times itself.

That 10 squared, 100.

10 cubed, 10 times 10 times 10,
so we've got 1,000.

And 10 to the fourth is
1, 2, 3, 4, we can just

keep adding those zeros.

So, we've got this log button,
fancy log button down here.

And so let's take the log of 1.

Log of 1 is 0.

Log of 10.

Log of 100.

Log of 1,000.

And you can almost
guess what the log of

10,000 is going to be.

So log of 1 is 0, 1, 2, 3, 4.

So what do you think
the log button does?

Well, what I want us to draw
our attention to is this

0 is the same as this 0.

This one, that one,
2 and 2, 3 and 3, 4, and 4.

So, what do we think
the log button does?

Well, if we take 10
to the 0 we get 1,

and if we take the
log of 1 it gives us

that exponent again.

And so the log is
undoing, it undoes

the exponent.

They are inverse functions

of exponents, or exponentials.

So here's what I mean.

If you have

b to the x equals a,
we can write that,

and this is called
exponential form.

We can write that as log
form, saying the log,

which is just another operation.

Base b

of a equals x.

And so this b, we call
the base of the log.

And this is what we're
taking the log of f.

And so the base of the log
and the base of the exponent

are the exact same thing, and
then the x and the a swap sides.

If you guys can know this and go
back and forth from this form,

you're going to go extremely
far with this logarithm concept.

To know that they
are the opposite of

each other like that.

And so what we're gonna
do is just practice

rewriting this like this.

So, rewrite these as logs.

10 to the third is a 1,000.

So, the log base 10,
because the base of the

exponent becomes the base of
the log, of 1,000 equals 3.

The log base 5,
the base of the

exponent of 625 equals 4.

Log base 2 of 1,024 equals 10.

And so now we wanna find x,
and so this is what we were

talking about before.

Log base 2 of 16 equals x.
Base of the exponent becomes

the base of the log, and now
the x is all by itself.

So if only we could
evaluate that.

Now, we didn't really
need logs for that,

and our calculator
can't do that outright,

but I'll show you how we
can adjust it for it.

Log base 2 of 20 equals
x, base of the log,

base of the exponent,
log base 2 of 50 equals x.

Let's get really good at
changing back and forth

between those two things.

So, something we need
to know, the common log.

Call it the common log and
that's what the log button

on our calculator is,
because notice the log

button on your calculator
doesn't have a number or

a base, it's because it
automatically does log base 10.

Because a ton of our numbers
are in the base 10 system,

we work in the base 10 system,
and so that's why it's on there.

And the natural log,
we already talked about

the natural number being e.

And so the natural log is
any log that has the base e.

And so, in both cases,
we don't write the bases,

and it's a little easier
to recognize, but you need

to notice what the base is.

So what if it doesn't
have base 10 or base e,

and we can't use the fancy
buttons on the calculator,

log and natural log?

Well, we use the
change of base formula.

And so we can change any
base, a, into base 10.

And we do the log of x
divided by the log of a.

Or you could use the
natural log if you wanted

to, whichever your preference.

They give you the
exact same answer.

Natural log of x,
natural log of a.

And the whole reason
behind it is because

you can really change it
to any base you want to.

Log of x divided by log
of a. This could be base

b and base b, as long as
they're the same base.

But most often,
we use log base b.

So let's utilize this.

Let's see what we can get.

Let's go back to our
other page and say

log base 16, log of
16 divided by log of 2.

And that's 4.

And so that's what
we got before.

Log 20 divided by log 2.

We said this one was
in between 4 and 5,

and so that one's
approximately 4.322.

So that if we go back to why
we were saying that, 2 to the

power of 4.322 is about 20.

And then finally,
log base 2 of 50.

Let's do the natural
log this time.

Natural log of 50 divided
by the natural log of 2.

And just to show you

that we get the
exact same thing,

no matter which
way we do it.

5.644.

All right, we'll come back
and talk more logarithms.