Calculus: Analytic Geometry and Calculus, with Vectors

9.1 Exponentials and logarithms 489

17 Persons who start fires with matchescan be interested in the fact that

fires can be started by rubbing sticks together. We shallsoon learn modern

ways of calculating approximations to exponentials and logarithms, butwe can

be interested in seeing that more primitive methods will work. Supposing that

a and x are positive numbers given in decimal form, and that 0 <x < 1, we

outline an old, old procedure for finding decimal approximationsto ax that was

used when all computations were made by hand and when letters A, B, C,

of the alphabet were used in place of xo,x,, x2,. To get started, let yo =

0, zo = 1, and observe that the inequalities

(1) yn_x5zn, ay" :- 5 : - 5

hold when n = 0. Supposing that n = 1, calculatex, and then all from the

formulas

(2) xn =yn-1 + zn_i 2 axn = `axn-1,

and let y,. and zn be two of the three numbersy,-i, x,,, zn_, chosen in such a way

that the inequalities (1) hold when n = 1. Repeating the process with n = 2

gives (2) and then (1) when n = 2, and the processcan be continued. Since

zn = yn + 1/2n, we can be sure that our estimates of ax will be good whenn is

large. To find

('/2 +

-/3)"-I

correct to 32 decimal places by this method (or by any other method) would be

quite a chore for an inexperienced person, but persons whorun modern computers

can enjoy making such computations. Primitive methods often work better

than fancier methods of limited applicability.

(^18) Persons possessing slide rules may wish to study them while reading
something that tells why they work. The C and D scales contain numbers from
1 to 10 and, when it is supposed that these scales have unit length (this unit is
usually about 10 inches), the number x lies logo x units from the left end. To
find the product X of two positive numbers A and B, we put them in the forms
A = 10-a and B = 1011b where m and n are integers and 1 < a < 10, 1 < b < 10
and use the fact that X = l0m+nx where x = ab. Noting that
(1) logo x = logo a + logo b,
we run along the D scale a distance loglo a to the number a and then, after setting
1 on the C scale opposite a, run along the C scale a distance logo b to the number b.
We have then gone the distance logos x from the left end of the D scale and hence
can read x on the D scale.
19 Persons possessing log-log slide rules may wish to see why and how the
esoteric LL scales are made. Suppose we wish to find the number y for which
y = b4 when b and q are given. We note that
(1)
and
loge y = q loge b
(2) logio loge y = logo loge b + logo q.