The Mechanical Engineer's Pocket-book

The minus sign is frequently written above the characteristic th log .002 3.30103. The characteristic only is negative, the decimal part mantissa, being always positive.

When a log consists of a negative index and a positive mantissa, it is us to write the negative sign over the index, or else to add 10 to the index, a to indicate the subtraction of 10 from the resulting logarithm. Thus log .2 = T.30103, and this may be written 9.30103 - 10. In tables of logarithmic sines, etc., the 10 is generally omitted, as bei understood.

Rules for use of the table of Logarithms.-To find ti log of any whole number.-For 1 to 100 inclusive the log is giv complete in the small table on page 129.

For 100 to 999 inclusive the decimal part of the log is given opposite t given number in the column headed 0 in the table (including the two figur to the left, making six figures). Prefix the characteristic, or index, 2.

For 1000 to 9999 inclusive: The last four figures of the log are four opposite the first three figures of the given number and in the vertic column headed with the fourth figure of the given number; prefix the tw figures under column 0, and the index, which is 3.

For numbers over 10,000 having five or more digits: Find the decimal pa of the log for the first four digits as above, multiply the difference figur in the last column by the remaining digit or digits, and divide by 10 if ther be only one digit more, by 100 if there be two more, and so on; add th quotient to the log of the first four digits and prefix the index, which is if there are five digits, 5 if there are six digits, and so on. The table of pro portional parts may be used, as shown below.

To find the log of a decimal fraction or of a whole number and a decimal.-First find the log of the quantity as if there were no decimal point, then prefix the index according to rule; the index is one less than the number of figures to the left of the decimal point. Required log of 3.141593.

Required the reciprocal of 3.141593.
Log of 3.141593, as found above...
Subtract decimal part from 0 gives.

0.4971498

0.5028502

Add 1 to the index, and changing sign of the index gives.. T.5028502 which is the log of 0.31831.

To find the fourth term of a proportion by logarithms. -Add the logarithms of the second and third terms, and from their sum subtract the logarithm of the first term.

When one logarithm is to be subtracted from another, it may be more convenient to convert the subtraction into an addition, which may be done by first subtracting the given logarithm from 10, adding the difference to the other logarithm, and afterwards rejecting the 10.

The difference between a given logarithm and 10 is called its arithmetical complement, or cologarithm.

To subtract one logarithm from another is the same as to add its complement and then reject 10 from the result. For a b = 10 − b + u – 10.

To work a proportion, then, by logarithms, add the complement of the logarithm of the first term to the logarithms of the second and third terms. The characteristic must afterwards be diminished by 10.

Example in logarithms with a negative index.—Solve by logarithms which means divide 526 by 1011 and raise the quotient

5262.45 1011

to the 2.45 power.