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.

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 2.56221, 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.

EXAMPLE II.

What are the logs. of 6543, 654.3, 65.43, 6.543, .6543, .06543 and .006543?

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

guished from absolute ones, by setting a negative sine over them, as above.

To find the Number of a given Logarithm.

Look for the given log. amongst the logs. from 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.

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 integers is one more than the given index as before.

The difference of these with three cyphers is for a

12)11000(916 Quotient.

108

20

12

80

Which quotient place after the first four figures found, and you have 3567916; and because the index is 4, the number will be 35679.16 required.

2. Required, the number answering to the log. 5.09901

The nearest log. to which is 3.09899, its No. 1256

Log. found 3.09899

Dividend 02000

Next log. 3.09934

35 Divisor 35)02000(57

175

250

245

5

Because the quotient consists of but two figures, prefix a cypher to it to make it three, and it is 057; which annexed to the first four found, is 1256057; and