CHAPTER III. LOGARITHMS. ART. 1.-In the expression Na"; "N" is the number obtained by raising "a" to the power of ""; keeping "a" fixed in value, and finding all the values of "N" obtained by performing the operation a: where "x" may be anything fractional or integral, results in a table of logarithms: the fixed, or constant, number is called the "Base," the power chosen the "Logarithm": thus a Logarithm is the index to which the Base must be raised to give the number. In common logs the base is always 10, hence the logs used in our problems represent the power to which 10 must be raised to equal the number. Logs are chiefly fractional, since very few numbers are exact powers of 10: the whole number, or integral part of the log., is called the index, and the decimal part the mantissa. It is the decimal part that is found on pages 2-16 in Norie's Tables, or any other nautical tables. The index is added by the computer, and is found by subtracting unity, "1," from the number of figures in the integral part of the quantity; this is evident from the following considerations: Hence the logs of all numbers between 1 and 10, those which only have one digit in the whole number part, fall between O and 1, and so are wholly decimal, having 0 for index. Again, the logs of all numbers between 100 and 1000, which are expressed by three digits in the whole number part, fall between 2 and 3, and so always have 2 for the index. The index of a wholly decimal number is negative, and is found by adding unity, "1," to the number of cyphers immediately after the decimal point, and before the first significant figure, and putting a minus sign (-) over it. D Logs of all numbers which are expressed by the same figures, but differ only in the position of the decimal point, have the same mantissa. A minus index is made positive by adding 10 to it. ART 2.-Finding the Logarithms of Numbers.-In Table, pp. 2-16, Norie, the first column is headed "No."; here are entered all numbers from 100 to 999. The other columns are headed 0 to 9; and the end one "Diff.": the first three figures of the number whose log is required are found under "No."; the fourth figure represents the column to be used; the number found in this column abreast the figures under "No." is the mantissa of log required. The index must be placed as indicated above. Should there be fewer than four figures in the number, add cyphers to make up four and take out mantissa, using index corresponding to original number. NOTE. The negative index is generally changed into a positive for Nautical Computations by adding 10, so that I becomes 9, 2 becomes 8, etc. When the number contains more than four figures, take out the log. of the first four; then find the correction for the remaining figures by multiplying the "Diff." by these figures, and cutting off as many to the right. The remaining quantity is the correction to be added. NOTE. There is no correction for O's at the end of a number. EXERCISES. 1. Find the Logarithms of the following numbers : ART. 3.-Finding the Numbers corresponding to Given Logarithms. -When searching for the number corresponding to a log., no account is taken of the index, the mantissa is looked out in the tables, and will be found to either correspond exactly with an entry there, or fall between two entries. In the event of exact coincidence, the figures in the "No." column and at the top of the column where the log. is found, give the first four figures of the Number required: the place of the decimal point depends on the Index, and is found by counting off from the left one more than the number of figures in the index. Should the index call for more than four digits in the Number, cyphers are added to make up the required quantity. Should the log. not agree, as is more often the case, the next less entry is taken from the tables and subtracted from the given log., cyphers are added to this difference and then divided by the number abreast in the "Diff." column. The figures in the dividend are extra ones to be placed to the right after the four figures taken from the tables corresponding to the less entry used. When the Index exceeds three, and more than four figures are required, the log. next less in the tables is taken out, and the remainder divided (as in Division of Decimals) by the "Diff.” gives the additional figures required. When the index is negative, and the number therefore a decimal, the position of the first figure (of value) is indicated by the index; thus, 1 shows it will occupy the first place after the decimal point, 2 requires the first figure (of value) in the second place, etc. NOTE. These index numbers when increased by 10 become 9, 8, 6. EXERCISES. 2. Find the numbers of which the following are the Logarithms : NOTE. Since logarithms are merely the powers of "10," which would make the numbers, to multiply quantities, we do as in Algebra, add the powers. To divide quantities, subtract the powers. To perform Involution, multiply the log. by the power. ART. 4. Multiplication by Logarithms. Rule.-Add together the logarithms of the numbers, and find the number corresponding to the sum of logarithms. Remark. It is evident that the product of a whole number by a whole number is a whole number, and the product of a decimal by a decimal is a decimal; but when a whole number and a decimal are multiplied together, the answer contains a whole number if the sum of indexes amounts to ten or more (in which case ten is rejected); but if the sum is less than ten, the answer is a decimal. |