logarithm 319

1

–

8

1

–

2

1

–

9

1

—

107

local maximum/local minimum See

MAXIMUM

/

MINIMUM

.

locus (plural, loci) A set of points satisfying some

specified condition is called a locus of points. For

example, the locus of all points in the plane at equal

distance from a given point is a circle, and the locus of

all points

EQUIDISTANT

from two given points Aand B

in the plane is a straight line perpendicular to the line

segment connecting Aand Bthrough its

MIDPOINT

.

Further, if the point Ais replaced by a circle, then the

locus of all points equidistant from Aand Bis a

HYPER

-

BOLA

if Blies outside the circle A, and an

ELLIPSE

if B

lies inside A. The locus of all points equidistant from a

line Aand a point Bis a

PARABOLA

.

If, in a system of C

ARTESIAN COORDINATES

, a locus

of points can be expressed in the form

{(x,y) : f(x,y) = 0}

then the equation f(x,y) = 0 is called the equation of the

locus. For example, x2+ y2– 25 = 0 is the equation of

a circle of radius 5 centered about the origin.

logarithm The power (

EXPONENT

) to which a num-

ber bmust be raised to obtain a given number Nis

called the base-blogarithm of N. That is, if bx= N,

then we write logbN= x, which we read as “the power

of bthat gives Nis x.” It is assumed that the number b

is positive. For example, one must raise the number 10

to a power of 2 to obtain the number 100, and so the

base-10 logarithm of 100 is 2:

log10 100 = 2

We also have:

log10 1,000 = 3 (the power of 10 that gives 1,000 is 3)

(the power of that gives is 3)

(the power of 3 that gives is –2)

log431 = 0 (the power of 43 that gives 1 is zero)

and

(the power of 6 that gives √

–

6 is 1/2)

Because a quantity bxis never negative, or zero, there is

no logarithm of a negative number or of zero.

Because logarithms are exponents, they obey the

same rules as exponents. For example, the multiplica-

tion rule for exponents reads bxby= bx+yindicating

that, upon multiplication, exponents add. This leads to

the rule of logarithms:

1. logb(N×M) = logbN+ logbM

(Precisely, if x= logbNand y= logbM, then we have:

bx= Nand by= M. Consequently, N×M= bx×by=

bx+y, which states that x+ yis the power of bthat gives

N×M. That is, logbN×M= x+ y= logbN+ logbM.)

Similarly, the exponent rule (bx)y= bxy leads to

the rule:

2. logb(Ny) = ylogbN

We also have the rules:

3. logb(bx) = x(The power of bthat gives bxis

indeed x.)

4. blogbx= x(Indeed, logbxis the power of bthat

gives x.)

5. logb1 = 0 (The power of bthat gives 1 is zero.)

Logarithms were invented by Scottish mathematician

J

OHN

N

APIER

(1550–1617) as a means to simplify arith-

metic calculations. For example, rule 1 shows that any

multiplication problem can be converted to the much

simpler operation of addition using logarithms. This dis-

covery was of great interest to scholars of the Renais-

sance, in particular astronomers, who were struggling

with problems requiring the manipulation of very large

numbers. Such computations were extremely tedious and

prone to many errors. Inspired by problems dealing with

the size of the Earth, Napier felt that working with

the number 107= 10,000,000 would be most helpful

to scientists, and he chose the number b= 1 –

as the base of his logarithms. Napier multiplied all

the quantities he worked with by 107to help avoid

the appearance of decimals. Today his logarithm of a

number Nwould be written .