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IF to a series of numbers in geometrical progression, whose common ratio is 10, and first term 1, we annex another series of numbers in arithmetical progression, whose first term is 0, and common difference 1; these latter numbers will be the logarithms of the former.
If several geometrical means be taken, and the like number of arithmetical ones, to the corresponding numbers, the latter will be the logarithms of the former.
The nature therefore of logarithms is such, that addition of them answers to the multiplication of their corresponding numbers; and subtraction to division; that is, when two numbers proposed are to be multiplied into each other, if we take the logarithms answering to those numbers, and add them together, the sum will be the logarithm answering to the number,
Again, when one number is proposed to be divided by another; if from the logarithm of the dividend, we subtract the logarithm of the divisor, the remainder shall be the logarithm of the quotient.
Most tables of logarithms contain the logarithms of all numbers from 1 to 10000, the column marked at the top N, is that in which you must find your number; in the same line with which in the adjacent column, is the logarithm of that number.
Required, the logarithm of 365.
And though most tables of logarithms run but to 10000, yet by them the log. of any number not exceeding 10,000,000 may be found, and on the contrary, the number to any such logarithm, thus,
1. Find the log. of the first four figures of the given number.
2. Take that log. from the log. of the number next following, and note their difference.
3. Multiply that difference by the remaining figures of the given number; and from the product cut off as many figures as remain in the given number, or as the given number is more than four (counting from the right to the left) as in decimals.
4. The whole number in the product, added to the first log. is the log. required; but the first figure, which is called the index, or characteristick must be changed; and always be one less than the number of figures in the logarithm.
Required, the logarithm of the number 3567194
The log. of 3567, which are the first four figures, is
The log. of the following or next number, viz. 3568,
*Cut off 3 figures, because 894 is three figures, and the product is
10.728 To which add the first log.
Their sum is
But because the given number consists of 7 figures, the index must be one less, which is 6; so the above index, 3, must be changed to 6, and we have 655240
Required, the log. of the number
Their sum is
Because the given number consists of 6 places, change the last index to 5, which is one less than the places in the given number; and you have 5.09901, the log. of 125607 required.
Because any number consisting of both integers and decimals, is equal to the quotient of the whole considered as an integer, divided by the denominator of the decimal part; and since by the nature of logarithms, subtraction in them answers the quotient of other numbers; therefore, it follows, that when a number is given, consisting of integers and decimals, we can find its log. thus ; find the log of the whole considered as one integer; then from that, take the log. of the denominator of the decimal part; or (which is the same thing) from the index of the log. of the whole considered as an integer, subtract a number less by one, than the number of places in the denominator of the fraction, and the remainder will be the log. required; or the index of the log. must be i less than the number of figures in the integer to which the decimal is annexed.
What is the log of the number 36.5 ?
Find the log. of 365, which is 2.5629; then because 10 is the denominator of the decimal part of the proposed number, and 1.00000 its log. therefore, from 2.56221, take 1.00000, and there remains 1.56229 the log. required.
Or because the whole number consists of two figures, the index of the log. must be one less, and is therefore 1.56229, as before.
What are the logs. of 6543, 654.3, 65.43, 6.543, .6543, .06543 and .006543 ? 6543
For the log. of a decimal fraction is the same as that of an integer; only the index is negative, and is so much less than 0. as the place of the decimal is re