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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.

EXAMPLE

Required, the logarithm of 365.

Answer, 2.56229.

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 inany figures as remain in the given number, or as the given number is more that 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 characteristic, must be changed ; and always be one less than the number of figures in the logarithm.

EXAMPLE I.

Required, the logarithm of the number 3567894

The log. of 3567, which are the first four figures is

3.55230 The log of the following or next number, viz. 3568, is

3.55242

Their difference,
Mult, by the remaining fig. viz.

12 894

Cut off 3 figures, because 894 is s figures, and the product is

10.728 To which add the first log.

3.55230

Their sum is

3.55240

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 6.55240 the log. of 3567894 required.

EXAMPLE II.

Required, the log. of the number 125607.
The log. of 1256 is

3.09899 The next log. following is

3,09934

Their difference is
Multiply by .07 the remaining figures

35 .07

Product
To which add the first log.

2.45 3.09899

Their sum is

3.09901 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 1 less, than the number of figures in the integer to which the decimal is annexed.

1 EXAMPLE I.
What is the log. of the number 36.5 ?

Find the log. of 365, which is 2.56229 : then because 10 is the denominator of the decimal part of the proposed number, and 1.00000 its log. therefore, from 2.56229, take 1.00000, and there remains 1.56229 the log. required.

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Look for the given log, amongst the logs. from

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.

EXAMPLE II. What are the log. of 6543, 654.3, 65.43, 6.543, .6543, .06543 and .006543 ? 6543

3.81578 654.3

2.81578 65.43

1.81578 6.543

0.81578 .6543

9.81578 .06543

8.81578 .006543

7.81578 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 removed from unity, and those indices may

be distingnished from absolute ones, by setting a negative sign over them, as above.

To find the number of a given Logarithm. 1000 to 10000 (not regarding the index or first figure) and if you find the exact log. you want, you have in the margin the required number. But if the index of the given log. be less than 3, cut off from the number found, as many figures as it is less; and the figures so cut off will be decimals, and the others integers. Or if the first figure or index, be greater than 3, add as many cyphers to the number found as it is more, and you have the number required.

EXAMPLES.

Find the numbers correspondent to the following logarithms. Given logarithms.

Numbers. 5.55230 Answer 356700. 4.55230

35670. 3.55230

3567. 2.55230

356.7 $1.55230

35.67 0.55230

3.567 9.55230

..3567 8.55230

.03567 7.55230

.003567, 8c.

But if the exact log. cannot be found in the table, and the number of figures required exceed four, then

1. Find as before (not regarding the index) the log. answering to the first four figures, but less than the given log

2. Take that from the given one, and if the remainder do not consist of two figures, prefix a cypher to it; and after these two figures annex three cyphers, so will you have five figures for a dividend.

3. Divide that by the difference between the log. found, and the next following, and if your quotient do not consist of three figures, prefix a cypher or cyphers to make it ; which three figures place after the first four found.

Then observe the index of the given log. which shews how many figures must be integers, and how many decimals; for the number of in

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