490 Exponential and logarithmic functions
The LL3 scale contains numbers from e to about 22,000. The distance from the
left end of the LL3 scale to a number y is loglo log, y. If we run a distance log10
log, b on the LL3 scale to hit b and then, with the aid of the C scale, run an addi-
tional distance loglo q, we go a total distance logio log, y and hence can read y on
the LL3 scale. We can be comforted by discovery that log-log slide rules give
the simple formulas 34 = 81 and 42.5 = 32 as well as others that are not so easily
verified. It is often particularly useful to know that if points b and f lie opposite
each other on the LL3 and D scales, then(3) logio log, b = logio fso log, b = f and b = ef. Thus natural exponentials and logarithms are easily
read, and the good approximations e3 = 20, log 20 = 3 are always available to
show us which way to read the scales. With the aid of this and similar informa-
tion about other scales, it is possible to both understand and use slide rules.
20 This is the promised development of the theory of ax for the case in which
x is rational. Let a be a positive number. It is the base of the exponential
function ax which we are about to define when x is an integer, then when x is the
reciprocal of a nonzero integer, and then when x is a rational number which is not
necessarily an integer. If n is a nonnegative integer, we define an and a-" by
the formulas
(1)1
an = 1'a'a'a'a a, a-" =1 a.a.a... a'where in each case 1 is multiplied by a exactly n times. If n = 0, no multiplica-
tions are involved and ao = 1. Clearly, a' = a, a -I = 1/a, a2 = a-2 =
etcetera, and a-" = 1/a" or a-na" = ao = 1. Counting numbers of times
by which 1 is multiplied or divided by a enables us to show that the laws of
exponents(2) azav = aiv, (az)v = axv
hold whenever x and y are nonnegative integers and then whenever x and y are
integers. Let n be a nonzero integer and let f(x) = x" when x > 0. Then
f'(x) = nx' ', so f is continuous and increasing when n > 0 and is continuous and
decreasing when n < 0. Moreover, f(x) -- 0 as x - 0 and f(x) --+ oo as x -> 00
when n > 0, and f(x) --> oo as x -* 0 and f(x) -). 0 as x oo when n < 0. In
each case these facts and the intermediate-value theorem 5.48 imply that there
is exactly one positive number h for which hn = a. We then define ailn by the
first of the formulas
(3) On = h, a = hn, (al/n)" = a, (an) 11n = a,remember that the first formula is equivalent to the second, and observe that the
last two formulas are correct. Supposing that X, p, and q are integers for which
Aq 0 0, we acquire ability to manipulate these things by proving the formula
(4) (a>' )11aq = (aP)llq = (ailq)" = (ail),q)XP
To prove this, let H be defined by ail'e = H, so that a = H'`q = (Hx)Q and
alle = H. Thus (4) will be true if(5) ((H19))1P)1/aq = ((Hq)P)11q = (Hx)P = H°.