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APPENDIX I.

ON THE LOGARITHMS OF NUMBERS, AND THE CONSTRUCTION OF

THE LOGARITHMIC TABLES OF NUMBERS.

1. Der. IF n = a*, x is called the logarithm of the number n to the base a; or, the logarithm of a number to a given base is that power to which the base must be raised to give the number.

If a logarithmic formula be generally true, whatever value the base may be of, the logarithms of the quantities involved will be written thus, log m, log n. But if the formula be true only when the logarithms are calculated to some particular base, (as 10, for instance), they will be written thus, log10 m, logion; or thus, llom, l10n.

If n = a*, and while a remains the same, successive values be given to n, and the corresponding values of x be registered, the tables SO

so formed are called tables of a system of logarithms to the base a.

It will hereafter be shewn, Art. 10, that a system of logarithms calculated to the base 10 is attended with peculiar advantages, and the tables of logarithms in common use are therefore calculated to that base.

2. Since if n = a*, x = lan; therefore, in all cases, n = Q = alan.

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=a,

.. m
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= a, and

ntan
... mem - niena

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man nem A true result, whatever be the value of a.

nelog n

= n

zlog m

..

Wherefore,

m

3. Required from tables of logarithms calculated to a given base, as , to form tables of logarithms to any other base, as 10.

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By Art. 2, 10

10 = €1,10
.. flon = 10/10" = (€1,103,2,0) = 61,10.1.0" ;
and equating the indices of €,

len=), 10.110n;
... 1.00 = 1.10.2en

.

Hence, the logarithms of any number n in two systems calculated to any bases, as 10 and , are connected by a constant multiplier, viz. ;

and therefore from tables

1,10

of logarithms calculated to a base €, we may form tables to the base 10.

n =

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lea

..len,

By writing a for 10 in this proof, we get 1, or, the constant multiplier connecting the logarithms of a number in the two systems whose bases are a and €, is

1

.

Ica

4. The logarithm of the product of any number of factors is equal to the sum of the logarithms of the several factors. For m.n.r... = ałam.ałam. ałam...

allam+len+ler+...) But m.n.r... = a(mnr...); ::: 1.(m.n.r...) = 1.m +1.n+lar + ... or, log (m.in. r...) = log m + log n + logr + ...

= a

Hence, if we find in the tables the number whose logarithm is the sum of the logarithms of the several factors, we obtain the product of those factors.

5. The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor.

1.) alem

= ałam-lan;

m

For a

= n

aan

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Hence, if we find in the tables the number whose logarithm is the logarithm of the dividend minus the logarithm of the divisor, we obtain the quotient.

6. The logarithm of the oth power of any number is equal to c times the logarithm of the number,-c being either whole or fractional. For ała(m") = mo

= m = (alemyo = q.1m;
... la (m“) = c.lam;
or log (mo) = c.log m.
Also
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Hence, the oth power of any given number is that number in the tables whose logarithm is c times the logarithm of the given number ; and the oth root of it is that number in the

is of the logarithm of the given

th

number.

7. From the last three articles it appears, that tables of logarithms enable us, particularly where the numbers are large, to perform the operations of multiplication, division, involution, and evolution, with greater ease than by the common arithmetical methods. The easier arithmetical operations of addition and subtraction cannot be performed by logarithms. 8. In the common, or Briggs', system of logarithms,

N where the base is 10, the logarithms of 10" N, and be determined from the logarithm of N. For 110 (10" X N) = 11,10+ 110N,

Art. 4. 11010 + 110N, Art. 6. = n + 110N.

Art. 2. Cor. 1.

Art. 5. = 11N - n.

10n, may

=n.

Thus we find in the pages of logarithms printed at the end of this Appendix, (p. 164.),

1.6 = 0.7781513 ;
.:: 11,60 = 1,010 + 1,6 = 1 + 0.7781513 = 17781513,

11.600 = 110(10%) + 1,06 = 2 + 0-7781513 = 2.7781513,
1106 = 11,6 – 11010 0:7781513 – 1,

110006 = 1,06 - 10(10%) = 0-7781513 – 3. The last two logarithms are thus written, 1.7781513, 3.7781513.

Der. The integral part of the logarithm is called the characteristic of the logarithm of the number; the decimal part is called the mantissa* of the significant digits of which the number is composed.

Thus, in 11,600 = 2.7781513, 2 is the characteristic of the logarithm of the number 600, 7781513 is the mantissa of the significant digit 6.

9. In the common system, to determine the characteristic of the logarithm of any given number.

If a number be between 1 and 10, its log. is between 0 and 1; .. its characteristic is 0. 10 and 100, 1 and 2 ;

1. 100 and 1000, 2 and 3;

2. 10^-1 and 101

n-1 and n;..

... n-1. Hence the characteristic of the logarithm of a number between 10"-1 and 10", (which has n integral places), is n - 1, or is less by one than the number of integral places which the number contains.

......

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*“MANTISSA ;" a handful thrown in over and above the exact weight; an overplus.

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