ELEMENTS OF SURVEYING. BOOK I. SECTION I. OF LOGARITHMS. 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 10° 1 101=10, = 103=1000 102 = 100, 10=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 numerical 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 (M× N); 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, 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`multiplied 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 |