WEBVTT

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- [Voiceover] We know
from previous videos,

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that the derivative with respect to X

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of the natural log of
X, is equal to 1 over X.

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What I want to do in this
video is use that knowledge

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that we've seen in other
videos to figure out

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what the derivative with respect to X is

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of a logarithm of an arbitrary base.

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So I'm just gonna call
that log, base A of X.

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So how do we figure this out?

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Well, the key thing is, is
what you might be familiar with

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from your algebra or your
pre calculus classes,

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which is having a change of base.

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So if I have some, I'll do it over here,

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log, base A of B, and
I wanted to change it

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to a different base, let's
say I wanna change it

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to base C, this is the same
thing as log, base C of B

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divided by log, base C of A.

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Log, base C of B, divided
by log, base C of A.

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This is a really useful
thing if you've never

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seen it before, you now have just seen it,

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this change of base, and we prove it

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in other videos on Khan Academy.

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But it's really useful
because, for example,

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your calculator has a log button.

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The log on your calculator
is log, base 10.

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So if you press 100 into your calculator

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and press log, you will get a 2 there.

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So whenever you just see log of 100,

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it's implicitly base 10,
and you also have a button

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for natural log, which is log, base E.

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Natural log of X is equal
to log, base E of X.

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But sometimes, you wanna
find all sorts of different

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base logarithms and this is how you do it.

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So if you're using your
calculator and you wanted to find

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what log, base 3 of 8 is, you would say,

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you would type in your
calculator log of 8 and log of 3.

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Or, let me write it this way, and log of 3

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where both of these
are implicitly base 10,

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and you'd get the same value if you did

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natural log of 8 divided
by natural log of 3.

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Which you might also
have on your calculator.

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And what we're gonna do
in this video is leverage

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the natural log because we
know what the derivative

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of the natural log is.

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So this derivative is the
same thing as the derivative

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with respect to X of.

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Well log, base A of X, can be rewritten as

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natural log of X over natural log of A.

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And now natural log of
A, that's just a number.

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I could rewrite this as,
let me write it this way.

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One over natural log of
A times natural log of X.

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And what's the derivative of that?

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We could just take the constant out.

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One over natural log of
A, that's just a number.

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So we're gonna get 1
over the natural log of A

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times the derivative with
respect to X of natural log of X.

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Of natural log of X.

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Which we already know is 1 over X.

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So this thing right
over here, is 1 over X.

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So what we get is 1 over
natural log of A times 1 over X.

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Which we could write as, 1
over natural log of A times X.

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Which is a really useful thing to know.

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So now, we could take
all sorts of derivatives.

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So if I were to tell you F of
X is equal to log, base 7 of X

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well now we can say well F
prime of X is going to be

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1 over the natural log of 7 times X.

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If we had a constant out
front, if we had for example,

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G of X.

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G of X is equal to negative
3 times log, base, I know.

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Log, base pi.

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Pi is a number.

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Log, base pi of X, well G
prime of X would be equal to

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1 over, oh.

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Let me be careful, I have
this constant out here.

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So it'd be negative 3 over,
it's just that negative 3,

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over the natural log of pi.

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This is the natural log of this number.

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Times X.

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So hopefully, that gives
you a hang of things.