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[INSTRUCTOR] All
right, logarithms.

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Logarithms, why we need
them is exactly this right

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here, what we're gonna see.

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If we're trying to solve for x

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

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okay, well, 2 times what?

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2 times itself how many times?

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Well, 2 times itself
four times is 16,

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and so x has gotta
be equal to 4.

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

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However, 2 to what power is 20?

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

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

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another 2, which is 32.

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And so 20 falls someplace
in between there.

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

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

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

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we need logarithms.

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Same deal with this one.

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

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

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another 2, which is 64.

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

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

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But what exactly we
need logarithms for?

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

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So 10 to the 0,
anything to the 0 power

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

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That 10 squared, 100.

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

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

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keep adding those zeros.

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

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And so let's take the log of 1.

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Log of 1 is 0.

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Log of 10.

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Log of 100.

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Log of 1,000.

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And you can almost
guess what the log of

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10,000 is going to be.

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

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So what do you think
the log button does?

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

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0 is the same as this 0.

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

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So, what do we think
the log button does?

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

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and if we take the
log of 1 it gives us

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that exponent again.

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And so the log is
undoing, it undoes

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the exponent.

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They are inverse functions

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of exponents, or exponentials.

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So here's what I mean.

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If you have

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

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and this is called
exponential form.

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

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which is just another operation.

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Base b

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of a equals x.

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

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

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

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

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

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

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To know that they
are the opposite of

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each other like that.

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And so what we're gonna
do is just practice

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rewriting this like this.

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So, rewrite these as logs.

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10 to the third is a 1,000.

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

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

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The log base 5,
the base of the

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exponent of 625 equals 4.

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Log base 2 of 1,024 equals 10.

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

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talking about before.

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

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

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So if only we could
evaluate that.

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

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and our calculator
can't do that outright,

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

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

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

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Let's get really good at
changing back and forth

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between those two things.

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

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

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on our calculator is,
because notice the log

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button on your calculator
doesn't have a number or

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

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

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

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And the natural log,
we already talked about

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the natural number being e.

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

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

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

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to notice what the base is.

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

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

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log and natural log?

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Well, we use the
change of base formula.

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

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

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Or you could use the
natural log if you wanted

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to, whichever your preference.

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They give you the
exact same answer.

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Natural log of x,
natural log of a.

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And the whole reason
behind it is because

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

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

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

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But most often,
we use log base b.

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So let's utilize this.

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Let's see what we can get.

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Let's go back to our
other page and say

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

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And that's 4.

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And so that's what
we got before.

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Log 20 divided by log 2.

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We said this one was
in between 4 and 5,

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and so that one's
approximately 4.322.

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

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power of 4.322 is about 20.

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And then finally,
log base 2 of 50.

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Let's do the natural
log this time.

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

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And just to show you

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that we get the
exact same thing,

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no matter which
way we do it.

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5.644.

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