ROGER BOWLEY: I'm going to be talking about logarithms,
logarithms to the base 10.
So for example, if I have a number, which is 100, I can
think of that as 10 squared, or 10 to the power 2.
So 2 is the logarithm of this number, 100, to the base 10.
1,000 is 10 to the 3.
This is the logarithm, the 3, and it corresponds to the
number of 0's there.
It is logarithm of 1,000 equals 3.
I can do 10 to another number.
So for example, 10 to the 1.6232 turns out to be 42.
That means that if I were to take 10, that means that the
log to the base 10 of 42 is 1.6232 to
four significant figures.
The reason it's useful is multiplication.
If I have two numbers and I want to multiply them
together, it's easy to make mistakes, because there are a
lot of little operations that have to be added up.
If you go through logarithms and you get a logarithm of a
number, such as 42, and you multiply it with another
number, such as 37, or 59, or 200, you get the logarithm of
that number, and you add these numbers together.
Let me show you an example.
37 times 59.
Now, I can write 37 as 10 to the 1.5682, and multiply that
by this number, which is 10 to the power 1.7709.
And when I multiply these numbers together, all I have
to do is add this number to that number to make this into
10 to the 3.3391.
BRADY HARAN: That's like a shortcut.
ROGER BOWLEY: Yes.
Instead of multiplying numbers, you're adding this to
that to get this exponent, and that number, it
turns out, is 2183.
Instead of having to write this out longhand as 37 times
59 and work it all out in gory detail, which will require 9
times 7 and 3 times 9, and then 5 times 7, and working it
out, and not making a mistake, all you have to do is
addition, which is much easier.
And if I were to do division, I would do subtraction, which
is much easier.
Addition and subtraction are much easier than
multiplication and division.
So this is a great simplification.
It makes your life much easier.
The catch is that you have to know these numbers.
37 is 10 to the power of this.
So where did I get that from?
Where did I get 59 is this number?
Well, the big secret, the thing that makes it possible
is that I have a table of these logarithms, and I look
up 37, this is actually 3.7, so in order to make it 37, I
have to add a 1 to it, and it's 1.5682.
So that's this number.
And then I look up 59.
And there's 59.
59.0, I go across , its 0.7709, but this is 5.9, so I
have to add a 1.
So these, I cheated, it's all in a table.
I can look it up.
When I was a lad at school, this is how we learned to do
multiplication.
I mean, we learned how to do long multiplication, but we
added these numbers together to get the answer quickly.
And it was the way of doing it, because there weren't
pocket calculators in my youth, because I'm so old.
This is all we had down mine.
So when I've got this number, I have to go to a different
table, which is called the table of antilogs.
Now, the 3 in the front of this means 1,000, so I just
have to look up 0.3391.
So I go down the table, there's 0.33.
And then you look across the top, and
there's a 9, 3391, 2183.
And then I look at the 1, which is this
thing, and add that.
And you see I get 2183.
And if there was a number there, I'd have to add that
last one, but it's actually 0.
So this is 2183, which turns out to be the
right answer exactly.
So you could get four significant figures by doing
this, which is reasonably accurate.
It's the speed, the reliability.
If these tables are accurate, you can get the numbers out.
BRADY HARAN: Is this just something that schoolboys and
schoolgirls would use, or were log tables important for
professionals?
ROGER BOWLEY: Oh, yes.
Log tables would be used by professionals who wanted to
get accuracy.
And they might not be just four significant figures, they
might be more, depending on the accuracy of the
calculations that you need to make.
And they made calculations very reliable, provided the
tables were reliable.
But the tables, to start with, weren't very reliable.
And that's really a complete story in itself, which I'd
like to tell you.
Well, now you're going to ask the question, where do these
numbers come from?
How did I make up tables of antilogarithms and logarithms?
And this was done by a man called John Napier.