WEBVTT
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PROFESSOR: OK.
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Hi.
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I thought I'd give a short
lecture about how logarithms
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are actually used.
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So a little bit practical.
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And also, it naturally
comes in, how quickly
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do functions grow?
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Which functions grow
faster than others?
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And I made a list of a bunch
of functions that
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we see all the time.
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Linear growth.
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Just, the function goes up
along the straight line.
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Proportional to x, linear could
have been a c times x,
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still linear.
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Here that's called polynomial
growth, like some power of x.
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Here is faster growth.
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We introduced e to the x, and
I'll take this chance to bring
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in 2 to the x and 10 to the x.
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Especially 10 to the x, because
that'll lead us to
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logarithms to base 10, and those
are handy in practice.
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So that's exponential growth.
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And here are some that
grow faster still.
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x factorial, n factorial grows
really fast. And n to the nth
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or x to the xth is a function
that grows still faster.
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And of course, we could
cook up a function the
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grew faster than that.
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X to the x to the x power would
really just take off.
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And we could find functions
that grow more slowly.
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But let's just take these
and let x be 1000.
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Just to have a kind of realistic
idea of how these
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compare when x is 1000.
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OK.
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So I'm skipping to c.
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So x will be 1000.
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10 cubed.
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Let me just write
it as 10 cubed.
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So x is going to be 1000.
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And because these are big
numbers, I'm going to write
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them as powers of 10.
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OK.
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so how about 1000 squared?
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10 cubed squared will
be 10 to the sixth.
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1000 cubed, we're up
to 10 to the ninth.
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And onwards.
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Like, this is where the
economists are working.
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The national debt is
in this range.
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OK.
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Now fortunately, it's
not in this range.
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2 to the thousandth power.
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And if I want to be able to
compare it, I'll write that
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approximately as 10 to--
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well, if it's 2 to the
thousandth power, it'll be 10
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to a smaller power.
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And 300 is pretty close for
2 to the thousandth.
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Then e to the thousandth, that's
going to be bigger than
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2. e is 2.7 et cetera.
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This is more like 10 to the--
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I think this is right--
about 434, maybe.
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And 10 to the thousandth--
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well, I can write
that right in.
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10 to the thousandth
when x is 1000.
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OK.
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So that's the one that
is exactly right.
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And also, I could write in 1000
to the thousandth power.
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What power of 10 will this be?
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10 to the what?
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1000 to the thousandth power,
I think, is 10 to the three
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thousandth.
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Why do I think that?
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Because 1000 itself is
10 times 10 times 10.
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Three of them, right?
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And then we do that 1000 times,
so we have a string of
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3000 10s multiplying
each other.
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And that's what 10 to the
three thousandth is.
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And you might wonder about
a thousand factorial.
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Let me make the rough
estimate.
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A big number in factorial,
order of magnitude, is
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something like, it doesn't grow
as fast as this, because
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this is x times x minus
1 times x minus 2.
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1000 times 999 times 998.
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So we're not repeating
1000 every time.
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And the difference--
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it turns out that this number
divided by this number, x to
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the x over e to the
x, is the right
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general picture for factorial.
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So that would be, if I divide 10
to the 3000 by 10 and this
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power, what do I do?
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In a division, I do a
subtraction of exponents,
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because I have that many fewer
10s multiplying each other.
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So I think it would be 3000,
but I don't want the full
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3000, because I take away e to
the thousandth, 434 of them.
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So that's about--
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2566 is close enough, anyway.
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OK.
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Giant numbers.
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Giant numbers.
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And of course you saw that I
didn't write it out with 1 and
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3000, or whatever, zeros.
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Hopeless.
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OK.
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In other words, it's the
exponent that gives me
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something I can really
work with.
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And the exponent is
the logarithm.
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That's what logarithms are.
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They are the exponents.
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And when they're the exponent
with a 10, I call 10 the base.
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And I'm speaking about
logarithms to the base 10.
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Can I just copy those
numbers again?
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And then I want to write their
logarithms. Because it's the
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logarithms that kind of remain
reasonable-looking numbers but
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tell you very nicely what's
growing fast.
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So let me write out again.
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10 cubed, 10 sixth, 10 to the
ninth is polynomial growth
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starting with the first power.
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Then I'll write down 10 to
the three hundredth,
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approximately.
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10 to the 434, I think,
is about right.
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And then 10 to the 1000.
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And then I had 10 to the 2566
as something, roughly 1000
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factorial, and then
10 to the 3000.
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OK.
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I just copied those
numbers again.
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And now I plan to take their
logarithms. I can see what's
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happening with logarithms.
The logarithm of 10
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to the ninth is--
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if the base is 10--
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the logarithm of 10 to the
ninth is the nine.
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This has logarithm 6.
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This has logarithm 3.
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So you see--
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well.
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If we took the logarithm of the
national debt, it wouldn't
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look too serious.
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It would just be up around
9 moving toward 10.
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But what I'm using it for
here is to get some
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reasonable way to see--
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300.
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Of course, that's big.
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For a logarithm, that's
a very big number.
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434, 1000.
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These are climbing up.
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2566 and 3000.
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OK.
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So these are the logs.
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Just to repeat.
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If I wanted this growth, this
list of functions by how fast
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they grow, where would
log x appear
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in my list of functions?
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It would be way at
the left end.
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Slower than x.
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Much slower than x.
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Log x grows very slowly,
as we see here.
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And then if you wanted one that
really grew slowly, it
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would be log of log x.
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That creeps along.
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Eventually gets to--
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passes any number.
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But x has to be enormous.
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And one more little comment
before I begin to use some
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things graphically.
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Because that's the other part
of this talk, is log--
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the graphs.
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Using logarithms in graphs.
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A little point.
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You might ask, what about
functions that decay?
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What would be the corresponding
functions here
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that decay?
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Let me write them here.
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Decay.
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By that I mean, headed
for 0 instead
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of headed for infinity.
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Well, 1 over x, 1 over x
squared, 1 over x cubed.
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Those functions go to
0 faster and faster.
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Now, what about these?
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The next list would
be 1 over--
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I'm dividing, but 1
over 2 to the x.
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1 over e to the x.
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Can I write that in
a better way?
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e to the minus x.
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1 over 10 to the x.
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Those are going to
0 like crazy.
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And of course, if I keep
going, even worse.
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So like, x to the minus x power
would be really small.
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So my point is just that we
have a scale here that not
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only gives us a handle of how
to deal with things that are
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growing very fast, but also
things that are going to 0
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very fast. The other, the
negative logarithms. The
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logarithms of these things would
be minus 3, minus 6,
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minus 9 and so on, if
I divide by one.
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Good.
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All right.
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So that suggests the idea.
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Now I want to introduce the
idea of a log scale.
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So I'm just going to think of
a usual straight line, on
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which we usually mark out 0,
1, 2, 3, minus 1, minus 2.
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But on this log scale, the
center point, the 0, I'm
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really graphing the logarithm
of x instead of x.
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That's the point.
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That in this log scale,
what I'm picturing
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along here will be--
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this number will be 10 to
the 0 power, which is 1.
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The next one will be 10.
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The next one will be 100.
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The next one will be 1000.
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So you see, within
this picture--
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on a graph that we could
draw and look at
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on a printed page--
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we can get big numbers by going
from the ordinary 1, 2,
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3 scale to the log scale, which
puts these points in
00:11:48.900 --> 00:11:49.430
this order.
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And let me put some
of the other ones.
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Now, what one point
goes there?
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1/10.
00:11:55.860 --> 00:12:00.640
Every time I go that far,
I'm multiplying by 10.
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When I go this way, I'm
dividing by 10.
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Up there, this is the number
1/10, which is the same as 10
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to the minus 1 power, right?
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Here is one hundredth.
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Here is one thousandth.
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And so on.
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So this log scale is able to
deal with very small numbers
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and very large numbers
in a reasonable way.
00:12:30.540 --> 00:12:35.530
And everybody sees the point
here that really, what it is
00:12:35.530 --> 00:12:39.170
is the logarithms.
So this is 0.
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This is 1, 2, 3, and so on.
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Minus 1, minus 2, minus 3.
00:12:46.500 --> 00:12:51.350
If I'm graphing, really, these
are the logarithms of x.
00:12:51.350 --> 00:12:55.470
And I'm doing logs to base 10
again, because that gives us
00:12:55.470 --> 00:12:57.630
nice numbers.
00:12:57.630 --> 00:12:57.705
OK.
00:12:57.705 --> 00:13:00.560
By the way, what's
that number?
00:13:00.560 --> 00:13:04.355
What's that number, halfway
between there and there?
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It's not halfway between 1 and
10 in the ordinary sense,
00:13:13.910 --> 00:13:15.760
which is whatever,
5 and a half.
00:13:15.760 --> 00:13:17.020
No way.
00:13:17.020 --> 00:13:20.520
Halfway between here is--
00:13:20.520 --> 00:13:21.580
you know what it will be?
00:13:21.580 --> 00:13:25.870
It'll be square root of 10.
00:13:25.870 --> 00:13:27.840
10 to the 1/2 power.
00:13:27.840 --> 00:13:33.720
The half is here.
00:13:33.720 --> 00:13:36.720
The log is a half, so
the number is the
00:13:36.720 --> 00:13:37.490
square root of 10.
00:13:37.490 --> 00:13:40.760
That's about 3, a little
more than 3.
00:13:40.760 --> 00:13:45.000
And what would be here, would
be 10 to the minus 1/2.
00:13:45.000 --> 00:13:47.450
1 over square root of 10.
00:13:47.450 --> 00:13:49.800
So you see that picture.
00:13:49.800 --> 00:13:53.840
Oh, I have another question,
before I use the scales.
00:13:53.840 --> 00:13:56.900
What if I like the powers
of 2 better?
00:13:56.900 --> 00:14:00.320
In many cases, we might
prefer powers of 2.
00:14:00.320 --> 00:14:04.370
Well, if I plotted
the numbers--
00:14:04.370 --> 00:14:07.620
I'm looking at this log scale.
00:14:07.620 --> 00:14:13.050
And suppose I plot the numbers
1, 2, 4, 8, whatever.
00:14:13.050 --> 00:14:14.720
16.
00:14:14.720 --> 00:14:16.520
What could you tell
me about those?
00:14:16.520 --> 00:14:17.890
Well, I know where 1 is.
00:14:17.890 --> 00:14:19.370
It's right there.
00:14:19.370 --> 00:14:20.780
That's a 1.
00:14:20.780 --> 00:14:22.900
Well, two would be a little
further over.
00:14:22.900 --> 00:14:26.590
Then 4, then 8 would come
before 10, and 16
00:14:26.590 --> 00:14:28.280
would come after 10.
00:14:28.280 --> 00:14:32.020
I pointed there, but 16
would not come there.
00:14:32.020 --> 00:14:37.390
16 would be a lot closer,
I think, in here.
00:14:37.390 --> 00:14:43.720
What's the deal with 1, 2, 4,
8, 16 on this log scale?
00:14:43.720 --> 00:14:45.970
They would be equally spaced.
00:14:48.790 --> 00:14:50.350
Of course, the spacing would be
00:14:50.350 --> 00:14:52.810
smaller than the 10 spacing.
00:14:52.810 --> 00:14:58.520
If every time I multiplied by
2, I go the same distance.
00:14:58.520 --> 00:15:01.500
After I'd done it
about 10 times--
00:15:01.500 --> 00:15:05.690
multiplied by 2 10 times-- so
that's 2 to the tenth power is
00:15:05.690 --> 00:15:07.060
close to 1000.
00:15:07.060 --> 00:15:11.445
So 10 powers of 2 would bring
me pretty near there.
00:15:11.445 --> 00:15:13.760
Anyway.
00:15:13.760 --> 00:15:15.830
And here's one more question.
00:15:15.830 --> 00:15:17.850
Where is 0?
00:15:17.850 --> 00:15:25.140
If my value that I wanted to
plot happened to be 0, where
00:15:25.140 --> 00:15:28.530
is it on this graph?
00:15:28.530 --> 00:15:30.670
It's not there.
00:15:30.670 --> 00:15:34.270
You can't plot 0
on a log scale.
00:15:34.270 --> 00:15:37.780
It's way down at the--
00:15:37.780 --> 00:15:41.620
you know, it's at the minus
infinity end of the graph.
00:15:41.620 --> 00:15:48.390
Infinity is up there at that
end, and 0 is down here.
00:15:48.390 --> 00:15:49.800
OK.
00:15:49.800 --> 00:15:50.330
Good.
00:15:50.330 --> 00:15:51.930
So can we use that log scale?
00:15:51.930 --> 00:15:53.780
How do we use that log scale?
00:15:53.780 --> 00:15:57.790
Let me give you an idea
for what use that
00:15:57.790 --> 00:15:59.710
log scale might be.
00:15:59.710 --> 00:16:01.520
Practical use.
00:16:01.520 --> 00:16:08.600
Suppose I know, or have reason
to believe, that my function
00:16:08.600 --> 00:16:17.580
might be of the form y is
something times x to the nth.
00:16:17.580 --> 00:16:22.030
I have some quantity y.
00:16:22.030 --> 00:16:25.880
The output when the
input is x.
00:16:25.880 --> 00:16:31.450
But I don't know these,
that number a.
00:16:31.450 --> 00:16:32.960
So I've done an experiment.
00:16:32.960 --> 00:16:35.930
And I would like to know
what is a, and
00:16:35.930 --> 00:16:37.420
especially, what is n?
00:16:39.950 --> 00:16:44.920
I would like to know how the
growth is progressing.
00:16:44.920 --> 00:16:50.220
And I'm just taking simple
growth law here.
00:16:50.220 --> 00:16:50.870
OK.
00:16:50.870 --> 00:16:51.960
I would graph it.
00:16:51.960 --> 00:16:54.710
I'd get a bunch of points, I
put them on a graph, and I
00:16:54.710 --> 00:16:56.410
look at the graph.
00:16:56.410 --> 00:17:00.080
Now if I just graph these
things, if I just graph that
00:17:00.080 --> 00:17:10.410
y, here is x and here's
y, suppose n is 1.5.
00:17:10.410 --> 00:17:13.680
Suppose my growth rate,
and this is very
00:17:13.680 --> 00:17:17.440
possible, is x to the 1.5.
00:17:20.410 --> 00:17:22.460
And a is some number--
00:17:22.460 --> 00:17:23.500
who knows.
00:17:23.500 --> 00:17:24.910
Could even be 1.
00:17:24.910 --> 00:17:26.160
Suppose a was 1.
00:17:30.670 --> 00:17:34.110
So then I'm graphing
y as x to the 1.5.
00:17:34.110 --> 00:17:36.060
What does that look like?
00:17:36.060 --> 00:17:39.890
Well, it looks like that.
00:17:39.890 --> 00:17:44.260
The problem is that if
the real growth--
00:17:44.260 --> 00:17:46.660
the real good relation--
00:17:46.660 --> 00:17:51.530
see, I would have a few
points that might be
00:17:51.530 --> 00:17:54.750
close to that curve.
00:17:54.750 --> 00:18:01.480
But if I'm looking that curve,
I frankly could not tell 1.5
00:18:01.480 --> 00:18:04.490
from 1.6 growth rate.
00:18:04.490 --> 00:18:07.600
The truth is, I couldn't
tell it from 2.
00:18:07.600 --> 00:18:11.290
I couldn't tell what the actual
growth rate is from my
00:18:11.290 --> 00:18:14.630
graph, which has a little error,
so I'm not too sure.
00:18:14.630 --> 00:18:18.450
And the point is x to the 1.5
and x to the 2 would be all--
00:18:18.450 --> 00:18:22.060
If I sketch the graph, it
would look like that.
00:18:22.060 --> 00:18:25.460
But go to the log scale.
00:18:25.460 --> 00:18:28.090
Go to a log log graph.
00:18:28.090 --> 00:18:31.220
So I'm going to take logs
of both sides, and
00:18:31.220 --> 00:18:33.890
look and plot that.
00:18:33.890 --> 00:18:37.180
So I take the logs of both
sides, so I take the log of my
00:18:37.180 --> 00:18:42.980
outputs y, and now this is a
product of that times that.
00:18:42.980 --> 00:18:45.030
What's the rule for
logarithms?
00:18:45.030 --> 00:18:49.590
Add logarithms. So this
would be log a plus
00:18:49.590 --> 00:18:53.990
log of x to the nth.
00:18:53.990 --> 00:18:58.370
But now what's the log
of x to the nth?
00:18:58.370 --> 00:19:00.570
Beautiful again.
00:19:00.570 --> 00:19:05.630
This is x times x
times x n times.
00:19:05.630 --> 00:19:08.000
At least of n is an integer.
00:19:08.000 --> 00:19:11.170
Think of it as x multiplied
by itself n times.
00:19:11.170 --> 00:19:15.400
When I take the logarithm,
I add n times.
00:19:15.400 --> 00:19:21.550
Log of x to the nth
is n log x.
00:19:21.550 --> 00:19:26.980
Now that, let me
graph that now.
00:19:26.980 --> 00:19:29.280
This is now a log picture.
00:19:29.280 --> 00:19:38.520
So I'm graphing log y against
log x, which was the whole
00:19:38.520 --> 00:19:43.450
point of my log scale, to
think of doing this.
00:19:43.450 --> 00:19:53.300
And what kind of a curve will
I see from this equation on
00:19:53.300 --> 00:19:54.550
this graph paper?
00:19:56.920 --> 00:19:58.170
A straight line.
00:20:00.490 --> 00:20:04.620
That is some constants
plus some slope.
00:20:04.620 --> 00:20:10.070
n will be the slope
times the x.
00:20:10.070 --> 00:20:17.360
It's like capital Y is capital
A plus n times
00:20:17.360 --> 00:20:18.900
capital X or something.
00:20:18.900 --> 00:20:20.910
But better for me to
write log, so we
00:20:20.910 --> 00:20:22.190
remember what it is.
00:20:22.190 --> 00:20:26.890
So on this paper, suppose--
00:20:26.890 --> 00:20:30.430
I did the example
x to the 1.5.
00:20:30.430 --> 00:20:31.850
OK.
00:20:31.850 --> 00:20:37.820
So in this example, a
is 1 and n is 1.5.
00:20:37.820 --> 00:20:40.090
So what would my points
look like here?
00:20:40.090 --> 00:20:46.260
Now remember, I should really
allow negative logarithms.
00:20:46.260 --> 00:20:50.450
Because this is the
point, right?
00:20:50.450 --> 00:20:52.270
This is x equals 1 here.
00:20:52.270 --> 00:20:54.565
The log is 0, but
the number is 1.
00:21:00.750 --> 00:21:00.890
Ha, OK.
00:21:00.890 --> 00:21:05.090
So when the log is 0, you
see, it's going to
00:21:05.090 --> 00:21:07.770
be a straight line.
00:21:07.770 --> 00:21:12.460
And actually, when I took a to
be 1, its logarithm will be 0.
00:21:12.460 --> 00:21:14.560
The line would go right
through there.
00:21:14.560 --> 00:21:17.230
It would have a slope
of 1 and 1/2.
00:21:17.230 --> 00:21:20.240
My points will be really
close to line.
00:21:20.240 --> 00:21:25.280
I measure out, if I go out a
distance 1, then I go up a
00:21:25.280 --> 00:21:26.458
distance 1.5.
00:21:26.458 --> 00:21:26.536
Right?
00:21:26.536 --> 00:21:27.786
Up 1.5.
00:21:32.880 --> 00:21:36.720
When I go across by 1 on
the log picture, it
00:21:36.720 --> 00:21:38.160
could be down here.
00:21:38.160 --> 00:21:42.540
My numbers could be
smaller or larger.
00:21:42.540 --> 00:21:43.470
A straight line.
00:21:43.470 --> 00:21:47.280
I can get out a ruler and
estimate the slope far more
00:21:47.280 --> 00:21:55.970
accurately than I could hear
with a lot more software.
00:21:55.970 --> 00:21:56.075
OK.
00:21:56.075 --> 00:22:03.500
So that's an important, very
important instance in which we
00:22:03.500 --> 00:22:06.180
wonder what the rate of
growth is, and the
00:22:06.180 --> 00:22:08.460
graph shows it to us.
00:22:08.460 --> 00:22:12.290
But just make a little point
that I've put some points
00:22:12.290 --> 00:22:18.420
here, like near a line, and
that raises another graph
00:22:18.420 --> 00:22:21.260
question of very great
importance.
00:22:21.260 --> 00:22:25.890
Suppose you have some
experiments that put points
00:22:25.890 --> 00:22:29.420
close to a line, but not
right on a line.
00:22:29.420 --> 00:22:32.700
You want to fit a line
close to them.
00:22:32.700 --> 00:22:37.950
You want to fit the best line
to the experimental points.
00:22:37.950 --> 00:22:40.830
How do you fit a
straight line?
00:22:40.830 --> 00:22:42.220
That's an important thing.
00:22:42.220 --> 00:22:48.040
And let me save that for a
future chance, because I want
00:22:48.040 --> 00:22:49.740
to tell you about it.
00:22:49.740 --> 00:22:52.460
The best, the standard
way is what's
00:22:52.460 --> 00:22:54.480
called the least squares.
00:22:54.480 --> 00:22:57.520
So least squares is a very
important application.
00:22:57.520 --> 00:23:02.970
And the best line, it turns out,
is a calculus problem.
00:23:02.970 --> 00:23:05.580
So for the moment let's
pretend they're
00:23:05.580 --> 00:23:07.570
right on the line.
00:23:07.570 --> 00:23:13.100
Its slope, which we easily find,
tells us this number.
00:23:13.100 --> 00:23:16.780
May I mention one
other behavior?
00:23:16.780 --> 00:23:19.190
So another possibility.
00:23:19.190 --> 00:23:24.320
If y is not growing
polynomially, but suppose y is
00:23:24.320 --> 00:23:28.560
growing exponentially--
00:23:28.560 --> 00:23:30.460
I'll just put it here,
because it's not
00:23:30.460 --> 00:23:32.290
going to be a big deal.
00:23:32.290 --> 00:23:35.410
y is some--
00:23:35.410 --> 00:23:41.250
call it b, e to the c x.
00:23:41.250 --> 00:23:42.830
So that's a different
type of growth.
00:23:42.830 --> 00:23:46.000
That's the big part of the
today's lecture, is to say,
00:23:46.000 --> 00:23:48.510
this is a quite different
growth.
00:23:48.510 --> 00:23:50.930
But it would be equally hard--
00:23:50.930 --> 00:23:52.480
or even harder--
00:23:52.480 --> 00:23:57.230
to find this growth rate c
from an ordinary graph.
00:23:57.230 --> 00:24:00.530
The graph would take off even
faster than this one.
00:24:00.530 --> 00:24:03.130
You couldn't see what's
happening.
00:24:03.130 --> 00:24:09.550
The good idea is, take
logarithms. But what do we
00:24:09.550 --> 00:24:10.610
want to do?
00:24:10.610 --> 00:24:12.950
We'll take the logarithm
of y--
00:24:12.950 --> 00:24:15.930
log y, as before--
00:24:15.930 --> 00:24:24.590
will be the log of B plus
the log of e to the cx.
00:24:24.590 --> 00:24:27.700
Oh, maybe I should have made
this 10 to the cx, just to
00:24:27.700 --> 00:24:29.640
make it all--
00:24:29.640 --> 00:24:31.850
instead of the e, I
could use the 10.
00:24:31.850 --> 00:24:34.040
Whatever.
00:24:34.040 --> 00:24:37.540
Because I've been talking about
logarithms to the base
00:24:37.540 --> 00:24:41.260
10, so let me use the
powers of 10 here.
00:24:41.260 --> 00:24:45.790
What's the logarithm
of 10 to the cx?
00:24:45.790 --> 00:24:50.190
When the base is 10, the
logarithm is the exponent.
00:24:50.190 --> 00:24:52.770
c times x.
00:24:52.770 --> 00:24:54.450
So what am I seeing
in this equation?
00:24:57.790 --> 00:25:01.370
That's an equation when I've
taken logarithms, my big
00:25:01.370 --> 00:25:03.950
numbers become reasonable.
00:25:03.950 --> 00:25:08.060
And also, very small numbers
become reasonable.
00:25:08.060 --> 00:25:11.990
And I get a straight
line again.
00:25:11.990 --> 00:25:14.170
I get a straight line.
00:25:14.170 --> 00:25:20.640
But it's not in this
log paper.
00:25:20.640 --> 00:25:25.290
The logarithm of y, the y-axis,
the vertical axis, is
00:25:25.290 --> 00:25:27.630
still log scale.
00:25:27.630 --> 00:25:30.130
But you see it's ordinary
x there now.
00:25:30.130 --> 00:25:33.440
So I don't use log
x for this one.
00:25:33.440 --> 00:25:35.510
Just ordinary x.
00:25:35.510 --> 00:25:37.780
It's semi log paper.
00:25:37.780 --> 00:25:41.060
Logarithm in the in the vertical
direction, ordinary
00:25:41.060 --> 00:25:42.970
in the x direction.
00:25:42.970 --> 00:25:44.720
OK.
00:25:44.720 --> 00:25:45.950
Good.
00:25:45.950 --> 00:25:52.380
Now I just want to add
one sort of example.
00:25:52.380 --> 00:25:56.083
Because it's quite important
and also quite practical.
00:25:56.083 --> 00:25:57.333
May I tell you about--
00:26:02.840 --> 00:26:06.520
Let me ask you the question,
and see if you get an idea.
00:26:06.520 --> 00:26:09.150
Because this is like
basic to calculus.
00:26:09.150 --> 00:26:12.880
Let me talk about--
00:26:12.880 --> 00:26:15.860
this e will stand for error.
00:26:15.860 --> 00:26:19.075
Error e.
00:26:19.075 --> 00:26:21.760
And what error am
I talking about?
00:26:21.760 --> 00:26:26.120
I'm talking about the error as
the difference between the
00:26:26.120 --> 00:26:28.490
derivative--
00:26:28.490 --> 00:26:33.920
I have some function f of x.
00:26:33.920 --> 00:26:36.000
And there's its derivative.
00:26:36.000 --> 00:26:44.420
And I compare that with
delta f over delta x.
00:26:44.420 --> 00:26:47.230
So what do I know?
00:26:47.230 --> 00:26:52.240
I know that as this is a
function of delta x, I'm
00:26:52.240 --> 00:26:58.410
comparing the instant slope
versus the average slope over
00:26:58.410 --> 00:27:02.040
a distance delta x.
00:27:02.040 --> 00:27:05.680
So it's not 0, right?
00:27:05.680 --> 00:27:10.040
This one is a finite movement.
00:27:10.040 --> 00:27:15.050
Delta x produces a finite
moment delta f.
00:27:15.050 --> 00:27:20.180
As delta x goes to 0, that
does approach this.
00:27:20.180 --> 00:27:22.720
So here's my question.
00:27:22.720 --> 00:27:27.990
My question is, this is
approximately some constant
00:27:27.990 --> 00:27:32.680
times delta x to some power.
00:27:32.680 --> 00:27:34.815
And my question is, what is n?
00:27:40.910 --> 00:27:41.940
How close?
00:27:41.940 --> 00:27:47.340
What's a rough estimate of how
near the delta f over delta x
00:27:47.340 --> 00:27:49.860
is to the actual derivative?
00:27:49.860 --> 00:27:51.180
OK.
00:27:51.180 --> 00:27:56.240
So I have to tell you what I
meant by delta f over delta x.
00:27:56.240 --> 00:28:05.040
I meant what you also meant, f
at x plus delta x minus f at x
00:28:05.040 --> 00:28:07.850
divided by delta x.
00:28:07.850 --> 00:28:10.720
In other words, that's
the familiar delta f.
00:28:10.720 --> 00:28:14.470
Moving forward from x, I would
call that a forward
00:28:14.470 --> 00:28:16.930
difference, a forward delta f.
00:28:16.930 --> 00:28:21.180
Because I'm starting at x, and
I think of delta x as moving
00:28:21.180 --> 00:28:22.940
me a little bit forward.
00:28:22.940 --> 00:28:26.240
So I get the delta f, I divide
by the delta x, and that's
00:28:26.240 --> 00:28:27.560
what this thing means.
00:28:27.560 --> 00:28:29.040
And do you know what n is?
00:28:33.620 --> 00:28:37.590
Let me connect it
to my pictures.
00:28:37.590 --> 00:28:42.345
If I tried to graph this,
I'd have a graph.
00:28:44.850 --> 00:28:45.690
You know.
00:28:45.690 --> 00:28:49.720
Here's my delta x
and here's my e.
00:28:49.720 --> 00:28:53.750
This difference says delta x
goes to 0, it goes to 0.
00:28:53.750 --> 00:28:57.370
You know, if delta x is
small, e is small.
00:28:57.370 --> 00:29:00.390
If I divide delta x by 10,
e divides by something.
00:29:04.060 --> 00:29:06.320
I don't even know if you
see it on the camera.
00:29:06.320 --> 00:29:08.800
The graph has gone into a--
00:29:08.800 --> 00:29:11.210
well, a black hole, or
a chalk hole, or a
00:29:11.210 --> 00:29:12.740
white hole, or something.
00:29:12.740 --> 00:29:16.550
It's just completely
invisible.
00:29:16.550 --> 00:29:18.940
I can't see the slope
of this thing.
00:29:21.570 --> 00:29:26.720
But if I did it on log log
paper, I'd see it clearly.
00:29:26.720 --> 00:29:30.420
And the answer would be 1.
00:29:30.420 --> 00:29:37.100
The error, the difference
between derivative and average
00:29:37.100 --> 00:29:43.210
slope, goes like delta
x to the first power.
00:29:43.210 --> 00:29:47.680
And then we can see later where
that 1 comes from, and
00:29:47.680 --> 00:29:49.800
we can see where that a is.
00:29:49.800 --> 00:29:52.690
It's all in Taylor series.
00:29:52.690 --> 00:29:57.150
But here's my practical point.
00:29:57.150 --> 00:30:00.820
There is a much better delta
f than this one.
00:30:00.820 --> 00:30:04.290
A much better delta
f over delta x.
00:30:04.290 --> 00:30:09.090
An average slope that's much
more accurate, and that in
00:30:09.090 --> 00:30:11.730
calculation I would
always use.
00:30:11.730 --> 00:30:14.900
And the trouble with this
one is, it's lopsided.
00:30:14.900 --> 00:30:15.920
It's one-sided.
00:30:15.920 --> 00:30:18.160
I only went forward.
00:30:18.160 --> 00:30:22.200
Or if delta x is negative,
I'm only going backwards.
00:30:22.200 --> 00:30:25.920
And it turns out that the
average of forward and
00:30:25.920 --> 00:30:31.200
backward is like centered
at difference.
00:30:31.200 --> 00:30:34.070
So let me tell you a center
difference. f at
00:30:34.070 --> 00:30:36.230
x plus delta x.
00:30:36.230 --> 00:30:40.690
So look a little forward, but
take the difference from
00:30:40.690 --> 00:30:41.940
looking a little backward.
00:30:44.410 --> 00:30:48.380
That would be my change in f.
00:30:48.380 --> 00:30:54.720
But now what do I divide by
to get a reasonable slope?
00:30:54.720 --> 00:31:00.050
Well, this is the change in f
going from minus delta x--
00:31:00.050 --> 00:31:03.370
delta x to the left of the point
to delta x to the right
00:31:03.370 --> 00:31:04.350
of the point.
00:31:04.350 --> 00:31:09.840
The real movement there in
the x-axis was a movement
00:31:09.840 --> 00:31:11.090
of two delta xs.
00:31:14.580 --> 00:31:17.200
So I would call this a
center difference.
00:31:17.200 --> 00:31:19.099
Can I write that word
"centered" down?
00:31:24.730 --> 00:31:29.810
And if I use that, which is a
lot smarter if I'm practically
00:31:29.810 --> 00:31:33.050
wanting to get pictures,
then what happens?
00:31:33.050 --> 00:31:37.350
So if this is now instead of
this, instead of choosing this
00:31:37.350 --> 00:31:45.200
lopsided, simple, familiar but
not that great difference, if
00:31:45.200 --> 00:31:53.710
I go for this one, the answer
is, n changes to 2.
00:31:53.710 --> 00:31:56.350
n is 2 for this one.
00:31:56.350 --> 00:32:03.960
The accuracy is way, way better
for center differences.
00:32:03.960 --> 00:32:09.510
And the point about the log
graphs is, if I plot those
00:32:09.510 --> 00:32:14.170
points on the graph I would see
that slope of 2 in the log
00:32:14.170 --> 00:32:17.460
log graph, it would be again--
00:32:17.460 --> 00:32:22.780
in ordinary graph, it would
become invisible
00:32:22.780 --> 00:32:24.690
as delta x got small.
00:32:24.690 --> 00:32:30.120
But on a log scale, I'd
see it perfectly.
00:32:30.120 --> 00:32:31.300
OK.
00:32:31.300 --> 00:32:35.650
Some practical uses of
logarithms. Now that we no
00:32:35.650 --> 00:32:39.550
longer use slide rules,
this is what we do.
00:32:39.550 --> 00:32:41.650
Thanks.
00:32:41.650 --> 00:32:43.410
NARRATOR: This has been
a production of MIT
00:32:43.410 --> 00:32:45.800
OpenCourseWare and
Gilbert Strang.
00:32:45.800 --> 00:32:48.080
Funding for this video was
provided by the Lord
00:32:48.080 --> 00:32:49.290
Foundation.
00:32:49.290 --> 00:32:52.420
To help OCW continue to provide
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00:32:52.420 --> 00:32:55.500
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