OF LOGARITHMIC TABLES

Excerpts

OF LOGARITHMIC TABLES
But the inventor of Logarithms was not a Peer, and should not be styled Baron Napier as is often done.
OF LOGARITHMIC TABLES
Logarithms of ordinary numbers may be defined to be numbers, so calculated from the ordinary numbers, that the sum of the logarithms of two numbers is the logarithm of their product.
OF LOGARITHMIC TABLES
Thus, with tables, the result of multiplication is calculated by adding logarithms, of division by subtracting logarithms, of raising to a power by multiplying a logarithm by a number, of extracting a root by dividing a logarithm by a number.
OF LOGARITHMIC TABLES
Equation (5) shews that the logarithm increases with the number: therefore it appears that the common logarithm of a number of one integral digit is a proper fraction; that of a number of $2$ digits is $1 + \text{a fraction}$; of $3$ digits $2 + \text{a fraction}$, and so on.
OF LOGARITHMIC TABLES
To avoid the use of negative numbers, the tabular logarithms of the circular functions are the common logarithms of these ratios increased by $10$: which must be remembered in using the tables in calculations.

OF LOGARITHMIC TABLES
\log m - \log n = \log(m ÷ n)
The difference of the logarithms of two numbers is the logarithm of their quotient.
OF LOGARITHMIC TABLES
\log m + \log n = \log (m × n)
The sum of the logarithms of two numbers is the logarithm of their product.
OF LOGARITHMIC TABLES
\log 1 = 0
The logarithm of 1 is zero.
OF LOGARITHMIC TABLES
\log n^x = x \log n
The logarithm of a power of n equals the exponent times the logarithm of n, for integer or fractional exponent x.
OF LOGARITHMIC TABLES
\log n^p = p \log n
For an integer p, the logarithm of n raised to the power p is p times the logarithm of n.
OF LOGARITHMIC TABLES
\log n^{\tfrac{p}{q}} = \frac{p}{q} × \log n
The logarithm of n raised to a fractional power p/q is p/q times the logarithm of n.
OF LOGARITHMIC TABLES
\log_a n = x, \text{ when } a^x = n
The logarithm of n to base a is the number x such that a raised to the power x equals n.
OF LOGARITHMIC TABLES
\log 10 = 1
The common logarithm of 10 is 1, since common logarithms have base 10.
OF LOGARITHMIC TABLES
\log (10^m × n) &= m + \log n
Multiplying a number by a power of 10 adds that integer power to its logarithm, changing only the characteristic.
OF LOGARITHMIC TABLES
\log (n \div 10^m) &= -m + \log n
Dividing a number by a power of 10 subtracts that integer power from its logarithm, changing only the characteristic.

No exercises in this chapter.