320 logarithmic function

a

Although logarithms are now understood as expo-

nents, Napier himself did not think of them in this way.

He developed his theory of logarithms geometrically,

thinking of them as a ratio of distances traveled by two

moving objects, one moving along a straight line at a

speed of 1 unit per second, and the other moving along

a line segment 1 unit long and with speed changing

according to its distance from the endpoint it is

approaching. Napier chose the name logarithm from

the Greek words logos for “ratio” and arithmos for

“number.” It was not until the end of the 17th century

that mathematicians recognized that logarithms were,

in fact, exponents.

After Napier published his work in 1614, English

mathematician H

ENRY

B

RIGGS

(1561–1630) suggested

to Napier that, like our number system, logarithms

should be based on the number 10. Napier agreed that

this would indeed simplify matters, and b= 10 was

then deemed the preferred base for logarithms. Base-10

logarithms are today called common logarithms or

Briggs’s logarithms. The common logarithm of Nis

simply denoted log Nor lg N.

In 1624 Briggs published tables listing values of

common logarithms for the numbers 1 to 20,000 and

90,000 to 100,000, inclusive. The values for the num-

bers 20,000–90,000 were completed after Briggs’s

death by Dutch mathematician Adriaan Vlacq.

Swiss watchmaker Jobst Bürgi, maker of astronom-

ical instruments, also conceived of logarithms to facili-

tate the multiplication of large numbers. However,

since Napier published his work first, the credit for

their discovery was not given to Bürgi.

A number with a given value for its logarithm is

called the antilogarithm, or antilog, of that value. The

base-10 antilog of a value xis 10x.

The common logarithms of the numbers 3.7, 370,

370,000, for example, differ by whole numbers: log 3.7

≈0.5682, log 370 = log(3.7 ×100) ≈0.5682 + 2 =

2.5682, and log 370,000 = log(3.7 ×105) ≈5.5682.

The decimal part of a logarithmic value is called the

mantissa, and the integer part is called the characteris-

tic of the logarithm. (For example, the three logarithms

above each have mantissa 0.5682 and characteristics 0,

2, and 5, respectively.) Logarithmic tables from the past

listed only the mantissas of numbers from 1 to 10. The

logarithm of any other number can then be computed

by adding the appropriate integer to represent the

power of 10 needed.

In

CALCULUS

it is convenient to work with loga-

rithms of base e. The number eis an irrational number

with value approximately 2.718281828… Logarithms

of base eare called natural logarithms, and a logarithm

of base eis denoted ln. Thus, ln(e3) = 3, for example.

If one is willing to work with complex numbers,

then it is possible to give meaning to the logarithm of a

negative number. For example, E

ULER

’

S FORMULA

tells

us that eiπ= –1. Consequently ln(–1) = iπ. Going fur-

ther, extending to logarithms of complex numbers, we

have, for instance, . In this

setting, all nonzero numbers—real and complex—have

logarithms.

See also

E

;

EXPONENTIAL FUNCTION

;

LOGARITHMIC

FUNCTION

;

LOGARITHMIC SCALE

;

SLIDE RULE

.

logarithmic function Any

CONTINUOUS FUNCTION

f(x),not identically zero, defined for positive values of

xwith the property that f(a· x) = f(a) + f(x) for all pos-

itive values aand x, is called a logarithmic function.

Such functions are said to “convert multiplication into

addition.” The series of observations below shows that

every logarithmic function is given by a

LOGARITHM

:

f(x) = logbxfor some positive base b.

1. All logarithmic functions f(x) satisfy f(1) = 0.

(This follows from the observation: f(1) = f(1.1) =

f(1) + f(1).)

2. All logarithmic functions satisfy .

Consequently, the logarithmic functions give both

positive and negative outputs.

(This follows from the observation:

.)

3. For any logarithmic function there is a number bfor

which f(b) = 1.

(We have f(1) = 0 and that it is possible to choose a value

asuch that f(a) is positive. One of the values f(a), f(a2) =

f(a) + f(a),f(a3) = f(a) + f(a) + f(a),… will be greater than

one. Thus it is possible to find a number cwith f(c) > 1.

By the

INTERMEDIATE

-

VALUE THEOREM

, there must be a

value bbetween 1 and c, so that f(b) = 1.)