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Derivatives of logarithms of different bases

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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.
Title:
Derivatives of logarithms of different bases
Description:

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Video Language:
English
Duration:
04:48

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