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exponent. It is evident that unity is excepted, since all the powers of 1 are 1.
A constant number being thus assumed, the series of exponents which must be assigned to this number in order to express the series of natural numbers, 1, 2, 3, &c., may be calculated and arranged in tables for use. Such a series of exponents constitutes a system of logarithms, and the constant number assumed is called the base of the system. Hence the following definition :
The logarithm of a number, in any system, is the exponent of the power to which the base of the system must be involved in order to produce that number.
59. The logarithm of a number is denoted by the abbreviation log., or simply by the letter l. or L.; and since a logarithm is nothing else than an exponent, we often use the notation
a representing the base of a system, b any number, and log. b the power to which a must be raised in order to produce b.
PROPERTIES OF LOGARITHMS IN GENERAL.
60. The logarithm of the product of two or more numbers is equal to the sum of the logarithms of those numbers.
For let b, c, d, &c. be any numbers, and a the base of any system of logarithms, then we have by the definition of logarithms
a log.b=b alog.c=c a log. d-d &c.
Multiply these equations together. The first members being powers of the same quantity are multiplied by adding their exponents (Art. 5,) therefore we have
a log.blog.c+ log.d+, =bcd... But we also have, by the definition of logarithms, a log. bcd... =bcd...
therefore we have the equation
a log. bcd...a log.blog.c+log.d+...
log. bed...=log. b+log. c+log.d+...
If, therefore, it is required to multiply two or more numbers together, we have only to take their logarithms from a table and add them together; the sum will be found in the table to be the logarithm of the required product.
61. The logarithm of the quotient of two numbers is equal to the difference of their logarithms.
For b and c being any two numbers, we have
But we also have
Dividing the first equation by the second, we have by Art. 6
whence the equation
=α log. b — log. c,
To divide one number by another, therefore, we take their logarithms from the table and subtract the log. of the divisor from the log. of the dividend; the remainder is found in the table to be the logarithm of the required quotient.
62. The logarithm of any power of a number is equal to the logarithm of that number multiplied by the exponent of the power. For b being any number, we have
Involve this equation to any power, as the nth. The first member is involved by multiplying the exponent log. b by n, according to Art. 8, and we have
a log.b" =b",
whence the equation
from which we have
log. b"-n log. b.
Therefore, to involve a given number to any power, we multiply the logarithm of the number by the exponent of the power; the product is the logarithm of the required power.
63. The logarithm of any root of a number is equal to the logarithm of that number divided by the index of the root.
For to extract the nth root of the equation
we divide the exponents of each member by n, according to Art. 9; we thus find
We also have, by the definition of logarithms,
a log. vb=√b.
n log. b
a log. ✓ba n
Therefore, to extract any root of a number, we divide the logarithm of the number by the index of the root; the quotient is the logarithm of the required root.
a=1. log. 1=0.
This follows also from Art. 60; for
64. The logarithm of unity is zero in all systems.
For a being the base of a system, we have by Art. 26, whatever is the value of a
log. bx1=log. b + log. 1,
and bx1 is b; so that this equation is simply
log.blog.b+ log. 1,
0 = log. 1.
65. The logarithm of the reciprocal of a number is the loga
rithm of that number taken negatively.
For b being any number, we have
a log. b=b,
whence we have
or by Art. 20
a log. b
(abc) = log
This also follows from Arts. 61 and 64, thus:
1 log. b
66. The logarithm of the base of a system is unity.
67. The student should make himself familiar with the mode of expressing algebraical quantities in logarithms. The following examples will be understood by referring to the preceding principles.
log. 1—log.b=0—log.blog. b.
4. log.ab=(log. a + log. b).
2. log. (amb"c...) = m log. a + n log. b+p log.c...
3. log. (176) )= m log. a + n log. b —p log. c.
= log. a+ log.blog.c―(log. d + log. e).
— (2 log. a + log. b + log. c— 2 log. d).
7. log. √a2—x2=[log. (a+x)+log. (a—x)].
8. log. ✔a2+x3. Here to apply logarithms we must resolve a2+x2 into factors. Let 2ax=y'; then we have a3+x3= a2+2ax+x2-2ax=(a+x)-y2=(a+x+y)(a+x—y);
The value of y is found by logarithms from y2=2ax,
11. log. (x2 — 1) — log. (x+1)+ log. (x — 1) = log. (x — 1)2.
12. log. 5+ log. 7- log. 9-log. 11+ log. x = = log.
13. 3 log.x+log.y—log.(x2+xy)—log.(y°+ xy)=log. („+y)*
* In some treatises, logarithms are defined to be "a series of numbers in arithmetical progression corresponding to a series of numbers in geometrical progression." But this is not a definition, but a consequence following from the definition; it is moreover calculated to confuse the student, who finds in his tables that the natural numbers are arranged in arithmetical progression, and not the logarithms.
If x is the logarithm of b in the system whose base is a, we have
ax-b, a3x-b3, a3x—b3, aax=b1, &c.,
whence we have the logarithms x, 2x, 3x, 4x, &c., in arith. prog. corresponding to the numbers b, b3, b3, b1, &c., in geom. prog. Again, if we multiply the equation ax-b by the equation a": obtain
=c, we shall
ax_b, ax+y=bc, ax+2y=bc3, ax+3y=bc3, &c.,
whence we have logs. x, x+y, x+2y, x+3y, &c., in arith. prog. and the numbers b, bc, bc2, bc3, &c., in geom. prog.