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[music] I'm going to approximate log 2 at
the natural log of 2 by starting at log

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of, of 1, which I know to be 0 and moving
over one quarter each time until after

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four steps I arrive at 2.
So here we go.

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First I'll try to approximate log of 1
plus a quarter.

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This will be approximately log of 1 plus a
quarter, which is how much I changed the

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input by, times, and I gotta approximate
the derivative somehow.

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I don't really want to approximate the
derivative of log at 1, and I don't really

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want to approximate the derivative of log
at 5 4ths.

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That will be an over and under estimate of
the derivative over the whole interval.

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So, I'm going to approximate the
derivative by evaluating the derivative in

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the middle of this region at 1 plus an
8th.

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Alright the derivative of log is 1 over,
so this is the derivative of log in the

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middle of the interval between 1 and 5/25.
Alright.

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So let's keep calculating here.
Log of 1 is zero.

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Plus a 4th times, what's 1 over 1 plus an
8th, 1 plus an 8th is 9 8ths, so that's a

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4th times 8 9ths.
Since I'm taking the reciprocal.

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And that is 2 9ths, so log of 5 4ths,
according to this is about 2 9ths.

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And I just keep on going right?
Now I want to approximate log Of one plus

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a quarter plus a quarter.
Well, that'll be about log of 1 plus a

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quarter, which is what I just calculated,
plus a quarter times, now I gotta write

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down the derivative, and, again, I'm going
to pick the derivative of the middle of

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this interval.
So, I'll evaluate the derivative at 1 plus

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3 8ths which is right in between 1 plus a
quarter and 1 plus 2 quarters.

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I approximated log of 1 plus a quarter to
be 2 nineth plus a 4th times, what's this?

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1 plus 3 8ths?
Well, that turns out to be 11 8ths but

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then I'm taking the reciprocal so I'll
write 8 11ths.

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So I've got 2 9ths plus 2 11ths.
That works out to 40 99ths.

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So that's approximately log of 1.5.
Let's keep on going.

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What's log of 1 plus 3 quarters.
Well, that should be about log of 1 plus 2

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quarters plus how much input changes to go
from here to here, which is a quarter

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times the derivative.
And now, where do I want to approximate

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the derivative?
I'll approximate it in the middle of the

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interval again.
So, it'll be 1 over 1 plus 5 8ths.

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Well what's this?
A log of 1 plus 2 quarters, log of 1.5, I

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just approximated that to be 40 99ths plus
a quarter times 1 plus 5 8ths, that's 13

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8ths.
The reciprocal is 8 13th.

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So this is 40 99ths plus 2 13ths which
this is going to be a, a huge number.

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It turns out 718 over 1,287.
And, that's approximately log of 1.75.

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Now, here we go.
What's log of 2?

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Well, that should be about log of 1.75.
So, 1 and 3 quarters plus 1 quarter, so

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that's how much the input change is to go
from here to here, times the derivative.

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Which I'm going to approximate as 1 over 1
plus 7 8th, right?

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I'm going to approximate the derivative in
the middle of this interval.

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And what does this turn out to be?
Well, I just approximated this.

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That was 718 over 1287 plus a quarter
times 1 plus 7 8th.

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That's 15 8th.
So 8 15th with the reciprocal.

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So, that's 718 over 1287 plus 2 15th which
turns out to be 4448 over 6435, which is

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approximately 0.691.
That is really great because log 2 Is

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actually closer to 0.693 but 0.691 is
awfully close to 0.693.

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And it's really fantastic that, you know,
just on paper, right?

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I'm able to get such a good approximation
to log 2.

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Let's use this same Euler method, or more
precisely, the midpoint method, to

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approximate log 3.
I use a little bit bigger step size here

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to save us some time.
I'm going to put this same gain to, to

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approximate log of 3.
Let's first do log 2.

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By writing log 2 as log 1 and I move from
1 to 2 by adding 1, so I'll add 1 times

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the derivative, and again I gotta pick 1
then I'll evaluate with the derivative.

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I don't want to evaluate the derivative
log at 1 or at 2, I want to do in between,

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so it would be 1 over 1 plus a half.
So log of 1 is 0 plus 1 plus a half is 3

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halves, but there's a reciprocal, so
that's 2 3rd.

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So this is telling me that log of 2 is
about 2 3rds.

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And I can do the same trick again with a
step size of 1, so log of 3 is log 2 plus

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1, which is how much I change by going
from 2 to 3 times the derivative, the

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approximation of the derivative somewhere,
i'll do it in between again so it'll be

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one over 2 plus a 1/2 right.
That's a point in between 2 and 3.

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And I approximated log 2 to be 2 3rds and
2 plus a half.

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That's 5 halves, by the reciprocal, that's
2 5ths.

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And 2 3rds plus 2 5ths, common denominator
of 15, is 16 15th which is about 1.07.

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How does this actually compare to log 3,
well log 3 is actually about 1.0986, so I

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mean seriously, 1.07 is not such a bad
guess, compared to 1.0986 blah, blah,

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blah, right?
In here I just used 2 steps and with a

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whole step size here of 1, right.
So this is even a coarser approximation

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than a, the thing I used to approximate
log 2.

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Now let's compute log 3 over log 2 or, if
you like, log 3 base 2.

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So log 3 over log 2 is about 1.584962501,
it keeps going, right?

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What you might notice here is that 19
12ths is 1.583.

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These are pretty close.
We can rephrase this without using

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logarithms at all.
So what I'm saying here is that log of 3

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over log 2 is about 19 12ths.
Well if this is true, that means that 12

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times log 3 is about 19 times log 2.
And my properties of logs, 12 times log 3

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is same as log 3 to the 12th and 19 log 2
is the same as log 2 to the 19th.

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Now if these logarithms are approximately
equal, you'd think then that 3 to the 12th

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would be approximately 2 to the 19th and
yes, sure enough 3 to the 12th is 531441,

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and 2 to the 19th is 524288.
And these numbers aren't so different.

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This turns out to be maybe more
significant than it seems at first.

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Let's play around with this some more.
These numbers 7, 12, 3 halves, 2, they're

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significant musically.
Alright?

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Two notes that are separated by a 5th are
actually in a ratio of 3 to 3 in terms of

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their frequencies.
Here, listen to two notes that are

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separated by a 5th.
Those two notes are a 5th apart.

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Now listen to two notes that are an octave
apart.

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Those two notes are an octave apart and
their frequency's in a ratio of two to

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one.
Now the deal is that at 12 fifths is about

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seven octaves.
Twelve fifths, that's like three halves,

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that's the ratio for a fifth to the
twelfth power is close to two to the

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seventh power, close to seven octaves.
Here, let's, let's listen to 12 5th in a

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row.
So that is 12 5th and now listen to seven

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octaves, starting at the same note.
Notice that last note, right?

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They're close to each other.
And that's really an audio proof that 3

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halves to the 12th power is almost two to
the seventh power.