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fourth power of any number, multiply its log. by 2, 3, or 4, according as the case may require; thus, to find the third power of 100, or (100)3, we multiply the log. of 100, which is 2, by 3, and the product 6, is the log. of (100)3, and the corresponding number to the log. 6, which is 1000000, is the third power of 100, or (100)3.

The converse of this is true; that is, that to extract the square root of any number, we divide the log. of that number by 2, and the number answering to the quotient is the root required. To extract the cube root of any number, we divide the log. of the number by 3, and the number answering to the quotient is the root required. Thus, to extract the square root of 10000, or (10000), the log. of 10000 is 4, which being divided by 2, gives 2, which is the log. of 100, the root required. In like manner, if we require to extract the cube root of 1000000, or (1000000), the log. of 1000000 is 6, which being divided by 3, gives 2, which is the log. of 100, the root required.

As the logs. of 1, 10, 100, 1000, &c. are 0, 1, 2, 3, &c. respectively, it is evident that the log. of any number falling between 1 and 10 will be 0 and some decimal parts; that of any number between 10 and 100, 1 and some decimal parts; of any number between 100 and 1000, 2 and some decimal parts; and so on for higher numbers. Hence, the index or characteristic of any log. is always 1 less than the number of figures in the integral part of the natural number.

Again, as a proper fraction is an expression arising from the division of the numerator by the denominator, and as this division is equal to the subtraction of their corresponding logs., it is obvious, that the log. of

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a proper fraction will have a negative index; thus the log. of, or .1, is found by subtracting the log of 100, which is 2, from the log. of 10, which is 1, the remainder is -1; in like manner, the log. of .01 is -2; that of .001, -3; that of .0001, -4; and so on for other decimals.

As the index or characteristic may be easily known by the computer, it is usually omitted in tables of logs. Most tables contain the logs. of the natural numbers from 1 to 10000, generally to six places of figures. In some tables, the logs. are continued to more decimal places.

To find the log. of any number, consisting of three places of figures; suppose 123.

Look in the left hand column for the number; then .08990 in the second column is the decimal part of its log., and the index is 2, as 123 consists of three integers; hence the log. of 123 is 2.08990.

To find the log, of any number consisting of four places of figures; suppose of 2157.

Look in the left hand column for the three first figures as before, that is 215; then under 7, at the top of the table, and in a horizontal line with 215, will be found 33385; and the index is 3, the integer consisting of four figures: therefore the log. of 2157 is 3.33385.

To find the log. of a figures; suppose 24 Find the log. of t last case, which de

then say, as 10 is to the difference, so is the fifth figure of the given number, to a fourth, which added to the less of the two logs., will give the log. sought.

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As 10 17 6: 10.2, the 4th number.

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The difference of the logs. is found in the right hand column of some tables, which saves the trouble of subtraction.

The above stating is founded on the supposition, that the differences of logs. are as the differences of the corresponding numbers.

Thus,

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Then 100 8:47 3.76, or 4 nearly.

Then .73743+4.73747; hence the log. of 546347 is 5.73747.

The calculator will soon perceive that it is unnecessary to state the numbers as above directed, it being sufficient to multiply the difference between the log. of the first four figures and the next log. greater, by the two remaining figures, and cutting off as many figures from the right hand of the product as are equal to the number of figures multiplied by.

To find the log. of a fraction; suppose 7.

Subtract the log. of the denominator from the log. of the numerator, and the remainder will be the log. of the fraction.

117

147

log. 2.06819
log. 2.16732

-1.90087 log. 17.

The fraction may be reduced to a decimal, and the log. found, as if whole numbers, except the index. If the significant figure be in the place of tenths, the index will be -1; if in the place of hundredths, it will be -2; if in the place of thousandths, it will be -3; and so on.

Thus, the log. of

.3754 is-1.57449

the log. of .03754 is-2.57449

the log. of .003754 is-3.57449, &c.

The log. of a mixed number is found as that of a whole number, except the index, which must be always 1 less than the number of places in the integral part. Thus, the log. of 59684 is 4.77586

And the log. of 59.684 is 1.77586

Also the log. of .59684 is-1.77586.

To find the number answering to any logarithm to four places of figures.

This process is only the converse of finding the log. answering to a given number. Therefore, look for the given log. among the columns containing the logs., the number in the left hand column will be the three first figures, and the figure at the top will be the fourth figure; the integral part is to be regulated by the index of the given log., as in the last case.

Thus, the number answering to the log. 2.32716 is 312.4

The number answering to the log. 4.35005 is 22390. If the given log. cannot be found exactly in the tables, take the difference of the logs. next greater and next less, and also the difference between the given log. and the next less; then say, as the first of these differences is to the second, so is 10 to the fifth figure of the required number.

If the number be required to six places of figures, make 100 the third term of the proportion, and the figures thus formed, when annexed to the number

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