Calculus: Analytic Geometry and Calculus, with Vectors

9.1 Exponentials and logarithms 485

and putting x = b in the first gives the second. Putting b = e and a= 10

gives the formula

(9.185) log x = (loglo x)(log 10) =loglo x

logio e

which, together with the estimates

(9.186) log 10 = 2.30258 50929 94045 68402

(9.187) log10 e = 0.43429 44819 03251 82765,

enables us to find log x with the aid ofa table of values of logio x. These

formulas are shunned when a log-log slide rule is available and gives
satisfactory accuracy and when a satisfactory table ofvalues of log x
is available. When decimal representations ofnumbers are used in cal-

culations, it is often necessary to know that each positive number y is
representable in the form
(9.188) y = 10'ax,

where n is an integer and 1 S x < 10, and that

(9.189) log. Y = n log. 10 + log.X.

This formula enables us to find log. y with the aid ofa table giving values
of lo& x when 1 < x 5 10. The formula works whenever a > 1. It is
simplest when a = 10 and log. 10 = 1; in thiscase it log. 10 is an integer
(the characteristic of the logarithm) and rules for finding itare sometimes
peddled without revelation of the fact that theyare identical with the
rules for finding the exponents in the representations

416.3 = 102 4.16
0.00004163 = 10-14.16
4.163 = 100 4.163.
Our derivations of formulas for derivatives and integrals of logarithms
and exponentials will come in the next section. Meanwhile, we close
this introductory section with some historical remarks. The first pub-
lished table of logarithms, the "Mirifici Logarithmorum Canonis "
by John Napier (1550-1617), appeared in 1614. The rare-book collection

of the University of Illinois contains this book and an astonishing col-
lection of old tables. To indicate that these books and their titles are

nontrivial, we cite the full title of the 1631 edition of the book of Napier.f

f Napier, John, "Logarithmicall Arithmetike, or tables of logarithmes for absolute
numbers from an unite to 100000; as also for sines, tangentes and secantes for every
minute of a quadrant: with a plaine description of their use in arithmetike, geometric,
geographic, astronomic, navigation, etc. These numbers were first invented by the most
excellent Iohn Neper, Baron of Marchiston, and the same were transformed, and the
foundation and use of them illustrated with his approbation by Henry Briggs, Sir Henry
Savils Professor of geometric in the Universite of Oxford. The uses whereof were v.ritten
in Latin by the author himselfe, and since his death published in English by diverse of his
friends according to his mind, for the benefit of such as understand not the Latin tongue,"
London, 1631, 819 pp.