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ELEMENTS OF SURVEYING.
1. The logarithm of a number is the exponent of the power to which it is necessary to raise a fixed number, in order to produce the first number.
This fixed number is called the base of the system, and may be any number except 1: in the common system. 10 is assumed as the base.
2. If we form those powers of 10, which are denoted by entire exponents, we shall have
101 = 10 103=1000
101=10000, &c., &c.,
From the above table, it is plain, that 0, 1, 2, 3, 4, &c., are respectively the logarithms of 1, 10, 100, 1000, 10000, &c.; we also see, that the logarithm of any number between 1 and 10, is greater than 0 and less than 1: thus, log 20.301030.
The logarithm of any number greater than 10, and less than 100, is greater than 1 and less than 2: thus,
The logarithm of any number greater than 100, and less than 1000, is greater than 2 and less than 3: thus,
If the above principles be extended to other numbers, it will appear, that the logarithm of any number, not an exact power of ten, is made up of two parts, an entire and a decimal part. The entire part is called the characteristic of the logarithm, and is always one less than the number of places of figures in the given number.
3. The principal use of logarithms, is to abridge nu merical computations.
Let M denote any number, and let its logarithm be denoted by m; also let N denote a second number whose logarithm is n; then, from the definition, we shall have, 10" = N (2).
10m = M (1)
Multiplying equations (1) and (2), member by member, we have,
10m+"= MXN or, m+n=log (MXN); hence,
The sum of the logarithms of any two numbers is equal to the logarithm of their product.
4. Dividing equation (1) by equation (2), member by member, we have,
Ñ or, m―n=log: hence,
The logarithm of the quotient of two numbers, is equal to the logarithm of the dividend diminished by the logarithm of the divisor.
5. Since the logarithm of 10 is 1, the logarithm of the product of any number by 10, will be greater by 1 than the logarithm of that number; also, the logarithm of the quotient of any number divided by 10, will be less by 1 than the logarithm of that number.
Similarly, it may be shown that if any number be muliplied by one hundred, the logarithm of the product will be greater by 2 than the logarithm of that number; and if any number be divided by one hundred, the logarithm of the quotient will be less by 2 than the logarithm of
From the above examples, we see, that in a number composed of an entire and decimal part, we may change the place of the decimal point without changing the deci mal part of the logarithm; but the characteristic is diminished by 1 for every place that the decimal point is removed to the left.
In the logarithm of a decimal, the characteristic becomes negative, and is numerically 1 greater than the number of ciphers immediately after the decimal point. The negative sign extends only to the characteristic, and is written over it, as in the examples given above.
TABLE OF LOGARITHMS.
6. A table of logarithms, is a table in which are written the logarithms of all numbers between 1 and some given number. The logarithms of all numbers between 1 and 10,000 are given in the annexed table. Since rules have been given for determining the characteristics of logarithms by simple inspection, it has not been deemed necessary to write them in the table, the decimal part only being given. The characteristic, however, is given for all numbers less than 100.
The left hand column of each page of the table, is the column of numbers, and is designated by the letter N; the logarithms of these numbers are placed opposite them on the same horizontal line. The last column on each page, headed D, shows the difference between the logarithms of two consecutive numbers. This difference is found by subtracting the logarithm under the column headed 4, from the one in the column headed 5 in the same horizontal line, and is nearly a mean of the differ