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 A and B
in the plane is a straight line perpendicular to the line
segment connecting A and B through its MIDPOINT.
Further, if the point A is replaced by a circle, then the
locus of all points equidistant from A and B is a HYPERBOLA if B lies outside the circle A, and an ELLIPSE if B
lies inside A. The locus of all points equidistant from a
line A and a point B is a PARABOLA.
If, in a system of CARTESIAN 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, x
2 + y
2 – 25 = 0 is the equation of
a circle of radius 5 centered about the origin.
logarithm The power (EXPONENT) to which a number b must be raised to obtain a given number N is
called the base-b logarithm of N. That is, if b
x = N,
then we write log b N = x, which we read as “the power
of b that gives N is 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:
log 10 100 = 2
We also have:
log 10 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)
log 43 1 = 0 (the power of 43 that gives 1 is zero)
and
(the power of 6 that gives √
–
6 is 1/2)
Because a quantity b
x is 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 multiplication rule for exponents reads b
x b
y = b
x+y indicating
that, upon multiplication, exponents add. This leads to
the rule of logarithms:
1. log b (N × M) = log b N + log b M
(Precisely, if x = log b N and y = log b M, then we have:
b
x = N and b
y = M. Consequently, N × M = b
x
× b
y =
b
x+y , which states that x + y is the power of b that gives
N × M. That is, log b N × M = x + y = log b N + log b M.)
Similarly, the exponent rule (b
x )
y = b
xy leads to
the rule:
2. log b (N
y ) = ylog b N
We also have the rules:
3. log b (b
x ) = x (The power of b that gives b
x is
indeed x.)
4. b
log b x = x (Indeed, log b x is the power of b that
gives x.)
5. log b 1 = 0 (The power of b that gives 1 is zero.)
Logarithms were invented by Scottish mathematician
JOHN NAPIER (1550–1617) as a means to simplify arithmetic calculations. For example, rule 1 shows that any
multiplication problem can be converted to the much
simpler operation of addition using logarithms. This discovery was of great interest to scholars of the Renaissance, 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 10
7 = 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 10
7 to help avoid
the appearance of decimals. Today his logarithm of a
number N would be written
.
10
10
7
1
1
10
7
7
log −
N
1
—
10
7
log 6 6
1
2
=
1
–
9
log 3
1
9
2
= −
1
–
8
1
–
2
log 1
2
1
8
3
=
logarithm 319
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 A and B
in the plane is a straight line perpendicular to the line
segment connecting A and B through its MIDPOINT.
Further, if the point A is replaced by a circle, then the
locus of all points equidistant from A and B is a HYPERBOLA if B lies outside the circle A, and an ELLIPSE if B
lies inside A. The locus of all points equidistant from a
line A and a point B is a PARABOLA.
If, in a system of CARTESIAN 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, x
2 + y
2 – 25 = 0 is the equation of
a circle of radius 5 centered about the origin.
logarithm The power (EXPONENT) to which a number b must be raised to obtain a given number N is
called the base-b logarithm of N. That is, if b
x = N,
then we write log b N = x, which we read as “the power
of b that gives N is 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:
log 10 100 = 2
We also have:
log 10 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)
log 43 1 = 0 (the power of 43 that gives 1 is zero)
and
(the power of 6 that gives √
–
6 is 1/2)
Because a quantity b
x is 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 multiplication rule for exponents reads b
x b
y = b
x+y indicating
that, upon multiplication, exponents add. This leads to
the rule of logarithms:
1. log b (N × M) = log b N + log b M
(Precisely, if x = log b N and y = log b M, then we have:
b
x = N and b
y = M. Consequently, N × M = b
x
× b
y =
b
x+y , which states that x + y is the power of b that gives
N × M. That is, log b N × M = x + y = log b N + log b M.)
Similarly, the exponent rule (b
x )
y = b
xy leads to
the rule:
2. log b (N
y ) = ylog b N
We also have the rules:
3. log b (b
x ) = x (The power of b that gives b
x is
indeed x.)
4. b
log b x = x (Indeed, log b x is the power of b that
gives x.)
5. log b 1 = 0 (The power of b that gives 1 is zero.)
Logarithms were invented by Scottish mathematician
JOHN NAPIER (1550–1617) as a means to simplify arithmetic calculations. For example, rule 1 shows that any
multiplication problem can be converted to the much
simpler operation of addition using logarithms. This discovery was of great interest to scholars of the Renaissance, 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 10
7 = 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 10
7 to help avoid
the appearance of decimals. Today his logarithm of a
number N would be written
.
10
10
7
1
1
10
7
7
log −
N
1
—
10
7
log 6 6
1
2
=
1
–
9
log 3
1
9
2
= −
1
–
8
1
–
2
log 1
2
1
8
3
=
logarithm 319