[music] I'm going to approximate log 2 at the natural log of 2 by starting at log
of, of 1, which I know to be 0 and moving over one quarter each time until after
four steps I arrive at 2. So here we go.
First I'll try to approximate log of 1 plus a quarter.
This will be approximately log of 1 plus a quarter, which is how much I changed the
input by, times, and I gotta approximate the derivative somehow.
I don't really want to approximate the derivative of log at 1, and I don't really
want to approximate the derivative of log at 5 4ths.
That will be an over and under estimate of the derivative over the whole interval.
So, I'm going to approximate the derivative by evaluating the derivative in
the middle of this region at 1 plus an 8th.
Alright the derivative of log is 1 over, so this is the derivative of log in the
middle of the interval between 1 and 5/25. Alright.
So let's keep calculating here. Log of 1 is zero.
Plus a 4th times, what's 1 over 1 plus an 8th, 1 plus an 8th is 9 8ths, so that's a
4th times 8 9ths. Since I'm taking the reciprocal.
And that is 2 9ths, so log of 5 4ths, according to this is about 2 9ths.
And I just keep on going right? Now I want to approximate log Of one plus
a quarter plus a quarter. Well, that'll be about log of 1 plus a
quarter, which is what I just calculated, plus a quarter times, now I gotta write
down the derivative, and, again, I'm going to pick the derivative of the middle of
this interval. So, I'll evaluate the derivative at 1 plus
3 8ths which is right in between 1 plus a quarter and 1 plus 2 quarters.
I approximated log of 1 plus a quarter to be 2 nineth plus a 4th times, what's this?
1 plus 3 8ths? Well, that turns out to be 11 8ths but
then I'm taking the reciprocal so I'll write 8 11ths.
So I've got 2 9ths plus 2 11ths. That works out to 40 99ths.
So that's approximately log of 1.5. Let's keep on going.
What's log of 1 plus 3 quarters. Well, that should be about log of 1 plus 2
quarters plus how much input changes to go from here to here, which is a quarter
times the derivative. And now, where do I want to approximate
the derivative? I'll approximate it in the middle of the
interval again. So, it'll be 1 over 1 plus 5 8ths.
Well what's this? A log of 1 plus 2 quarters, log of 1.5, I
just approximated that to be 40 99ths plus a quarter times 1 plus 5 8ths, that's 13
8ths. The reciprocal is 8 13th.
So this is 40 99ths plus 2 13ths which this is going to be a, a huge number.
It turns out 718 over 1,287. And, that's approximately log of 1.75.
Now, here we go. What's log of 2?
Well, that should be about log of 1.75. So, 1 and 3 quarters plus 1 quarter, so
that's how much the input change is to go from here to here, times the derivative.
Which I'm going to approximate as 1 over 1 plus 7 8th, right?
I'm going to approximate the derivative in the middle of this interval.
And what does this turn out to be? Well, I just approximated this.
That was 718 over 1287 plus a quarter times 1 plus 7 8th.
That's 15 8th. So 8 15th with the reciprocal.
So, that's 718 over 1287 plus 2 15th which turns out to be 4448 over 6435, which is
approximately 0.691. That is really great because log 2 Is
actually closer to 0.693 but 0.691 is awfully close to 0.693.
And it's really fantastic that, you know, just on paper, right?
I'm able to get such a good approximation to log 2.
Let's use this same Euler method, or more precisely, the midpoint method, to
approximate log 3. I use a little bit bigger step size here
to save us some time. I'm going to put this same gain to, to
approximate log of 3. Let's first do log 2.
By writing log 2 as log 1 and I move from 1 to 2 by adding 1, so I'll add 1 times
the derivative, and again I gotta pick 1 then I'll evaluate with the derivative.
I don't want to evaluate the derivative log at 1 or at 2, I want to do in between,
so it would be 1 over 1 plus a half. So log of 1 is 0 plus 1 plus a half is 3
halves, but there's a reciprocal, so that's 2 3rd.
So this is telling me that log of 2 is about 2 3rds.
And I can do the same trick again with a step size of 1, so log of 3 is log 2 plus
1, which is how much I change by going from 2 to 3 times the derivative, the
approximation of the derivative somewhere, i'll do it in between again so it'll be
one over 2 plus a 1/2 right. That's a point in between 2 and 3.
And I approximated log 2 to be 2 3rds and 2 plus a half.
That's 5 halves, by the reciprocal, that's 2 5ths.
And 2 3rds plus 2 5ths, common denominator of 15, is 16 15th which is about 1.07.
How does this actually compare to log 3, well log 3 is actually about 1.0986, so I
mean seriously, 1.07 is not such a bad guess, compared to 1.0986 blah, blah,
blah, right? In here I just used 2 steps and with a
whole step size here of 1, right. So this is even a coarser approximation
than a, the thing I used to approximate log 2.
Now let's compute log 3 over log 2 or, if you like, log 3 base 2.
So log 3 over log 2 is about 1.584962501, it keeps going, right?
What you might notice here is that 19 12ths is 1.583.
These are pretty close. We can rephrase this without using
logarithms at all. So what I'm saying here is that log of 3
over log 2 is about 19 12ths. Well if this is true, that means that 12
times log 3 is about 19 times log 2. And my properties of logs, 12 times log 3
is same as log 3 to the 12th and 19 log 2 is the same as log 2 to the 19th.
Now if these logarithms are approximately equal, you'd think then that 3 to the 12th
would be approximately 2 to the 19th and yes, sure enough 3 to the 12th is 531441,
and 2 to the 19th is 524288. And these numbers aren't so different.
This turns out to be maybe more significant than it seems at first.
Let's play around with this some more. These numbers 7, 12, 3 halves, 2, they're
significant musically. Alright?
Two notes that are separated by a 5th are actually in a ratio of 3 to 3 in terms of
their frequencies. Here, listen to two notes that are
separated by a 5th. Those two notes are a 5th apart.
Now listen to two notes that are an octave apart.
Those two notes are an octave apart and their frequency's in a ratio of two to
one. Now the deal is that at 12 fifths is about
seven octaves. Twelve fifths, that's like three halves,
that's the ratio for a fifth to the twelfth power is close to two to the
seventh power, close to seven octaves. Here, let's, let's listen to 12 5th in a
row. So that is 12 5th and now listen to seven
octaves, starting at the same note. Notice that last note, right?
They're close to each other. And that's really an audio proof that 3
halves to the 12th power is almost two to the seventh power.