Mathematical Excalibur, Volume 7, Number 1 (English translations of Chinese articles)

C o n s t r u c t i o n of Logarithm Tablc LEE R’zng Lung

I11 this modern age of computers, to find a logarithmic number like l n 2 requires only the touch of a

finger. Have you ever wondered, before tlie appearance of calculators, how tlie thick logarithm tables were constructed with precisions? Of course, historically there appeared many methods of constructions, each has its advantages, each has its limits. Below we will discuss one such inetliod systematically. It only uses a bit of basic calculus skills and yields an efficient, way of constructing a logarithm table to any precision.

First note that since ln(xy) = l n x

+

In y, we only need to find the logaritlim of prime numbers p to get the logarithm of the other positive integers. From the differentiation of ln(1

+

t )

and the geometric series formula, we get

By the fundamental theorem of calculus (that is the inverse relations of differelltiation and integration), we get

for all positive integer n. Let us now estimate the integral remainder term. For 1x1

<

1, we have

From this we can see that as n increases, the absolute ralue of tlie remainder term will go to 0. In other x2 x 3 2 4 X n

words, for n sufficiently large, the difference between ln jl

+

x) and x - -

+

- -

-

+

. .

.

+

(-l)”-’-

2 3 4 n

will become arbitrarily sinall. Hence we may represent this in the following way:

x’ x3 24

2 3 4

ln(1

+

x) = x

-

-

+

-

- -

+

_{. . . (1x1}

_<

_1).

Thus by a choice of TI we may ignore tlie difference. Replacing x by --G and subtracting the two equation,

we have the following equation

l + x 23 x 5

In(=) = l n ( l + x ) - ~ n ( ~ - x) = 2 ( c + - 3

+ - + .

5

Unfortunately, if we substitute x = P - - to get -

=

p, equation

(*)

will not be able to produce 2 9 - 1 14

l n p efficienrly. For example, take p = 39, then L = ~ = - and compute even to the 100th term

1 9 + 1 1.5’

P + 1 1--3:

~. , ~ ~.

2x’99

-

1 nn 1.1 x lo-’, the value of l n p is still not determined precisely to the 8th decimal place (more rig-

I Y J

orously, we should have used the remainder term of the equation (*) to estimate the difference, but here we just want to know the difference approximately); for another example, take p = 113, then

x

=

-

and 2,199

- M 3 I.:

56 57 the precision of l n p is even worse. However, if we use x =

-

then

199 2 p 2 - 1

For prime p

>

2 , the coininon priine divisors of ( p

+

1) and (p

-

1) must be less than p . So if we have coinputed the logarithm of all priine numbers less than p, then we can use the above equatioii to coinpute the value of lnp:

2 l n p = lri(’=) + l n ( p + 1) +l ii ( p - 1).

Now the term 111

(i

-

:)

can be computed efficiently because tlie absolute value of z chosen will be small. so just compute to - M 3 :=: 10-l’ we can obtain tell For example, when p = 29, x =

decimal place accuracy.

l + Z 2 x 5 1

-

- - 1 2 . 2 9 ? - 1 1681 ’ 5

From the discussion above, suppose we now want to coiistruct a 8 decimal place logarithm table. Then we can compute the logarithm of 2 , 3 , 5 , 7, 11,13,.

.

. in order and the logarithm of the numbers can be obtained from the logarithm of the numbers preceding them. From this we see that in the begiiiiiing we need to compute In 2 to a high precisioii:

(;)?I

-)

=

0.6931471805589..

..

($)3

(;I5

1n2

=

ln(-+) M 2 ( ;

+

-

+

-

+ .

.

. +

1 1 - - 1 + , 3

5

21

This and the actual answer In2 = 0.693147180559945.. . agree t o 11 decimal places. Then we compute hi 3. 1 - we have Taking

x

= ~ - - 1 2 . 3 ” l 17’ I 1 3 1 - - 1

(n)

₊

CAI5

(&I7

-

+

T)

= 0.117783035654504.. . . In(+) 1 + f i M 2 ( F

+

5

2 ( 3 9

Note that since - M 1.9 :x: lo-”, we may ignore this if only 8 decimal place accuracy is desired. So 9

1

l n 3 M 5(0.11578303565..

.+

l n 4 - t 1112) = 1.098612288635.. ..

1

This and the actual answer In3 = 1.09861228866811.

..

agree t o 10 decimal places. We will leave the coinputations of ln5, In7 and so on to t h e reader and let them compare with the results given by calculators.

Looking back at the above very clever computatioii method, everyone should respect the inathematiciaiis in the past for their creativity and instiiict in the relations of numbers and computations!

Rcfcrcnccs:

Chapter 3 of Professor Wu-Yi Hsiang’s Notes for his Lectures on Aiialysis available at http://ihome.ust .lik/-maIung/391. html