We add 2 to 1.8055 rather than 1, because 1.68 is nearer 2 than 1. In general we take the nearest integer.

By Theorem II, § 209, (log 275.4 × 1.463) = 2.6051

213. Antilogarithms. We can now find the product or quotient of two numbers if we are able to find the number that corresponds to a given logarithm.

For this process we have the following

RULE. If the mantissa is found exactly in the table, the first two figures of the corresponding number are found in the column N of the same row, while the third figure of the number is found at the top of the column in which the mantissa is found.

Place the decimal point so that the rules in § 210 are fulfilled.

EXERCISES

Find the antilogarithms of the following:

1. 3.7419.

Solution: We find the mantissa .7419 in the row which has 55 in column N. The column in which .7419 is found has 2 at the top. Thus the significant figures of the antilogarithm are 552. Since the characteristic is 3, we must by the rule in § 210 have four figures to the left of the point. Thus the number sought is 5520.

If the mantissa of the given logarithm is between two mantissas in the table, we may find the antilogarithm by the following RULE. Write the number of three figures corresponding to the lesser of the two mantissas between which is the given mantissa. Subtract this mantissa from the given mantissa, and divide this number by the tabular difference to one decimal place.

Annex this figure to the three already found, and place the decimal point as the rules in § 210 require.

It should be kept in mind that we may always add and subtract any integer to a logarithm. This is useful in two cases:

First. When we wish to subtract a larger logarithm from a smaller;

Second. When we wish to divide a logarithm by an integer that is not exactly contained in the characteristic.

Both these processes are illustrated in exercise 2 (1) following.

EXERCISES

1. Find the antilogarithms of the following:

(a) 2.3469.

Solution: The mantissa 3469 is between 3464 and 3483. Hence D = 19. The mantissa 3464 corresponds to 222. To find the fourth significant figure of the antilogarithm, divide 3469 - 3964 5 by D 19. Since 5 ÷ 19 = 2.6, we annex 3 to 222. Hence the antilogarithm = 222.3.

=

We write 2+.8414 in the form 2.8414 to save space and at the same time to recall the fact that the mantissa is positive.

10 by 3, we

Since in the subtraction in this problem we have to subtract 7 from 3, we add and subtract 10 to the minuend to avoid a negative logarithm. Since in the division by 3 we would have a remainder in dividing add and subtract 20 so that 3 may be exactly contained in 30, the negative part of the logarithm.

214. Cologarithms. The logarithm of the reciprocal of a number is called its cologarithm. When a computation is to be made

in which several numbers occur in the denominator of a fraction, the subtraction of logarithms is conveniently avoided by the use of cologarithms. By our definition we have

colog 25 = log log 1 - log 25,

log 1

=

=

Theorem III, § 209

Thus in dividing a number by 25 we may subtract the logarithm of 25, or what amounts to the same thing, add the logarithm of 2, which is by definition the cologarithm of 25.

RULE. The cologarithm of any number is found by subtracting its logarithm from 10

10.

In the process of division subtracting the logarithm of a number and adding its cologarithm are equivalent operations.

215. Change of base. We have seen that the logarithm of a number for the base 10 may be found to four decimal places in our tables. It is occasionally necessary to find the logarithm of a number for a base different from 10. For the sake of generality, we assume that the logarithms of all numbers for a base bare computed. We seek a means of finding the logarithm of any number, as x, for the base c; that is, we seek to express log, in terms of logarithms for the base b.

Suppose

log.xz, that is, c2 = x.

Take the logarithm of this equation for the base b, and we have

This number M does not

but only on the two bases.

(1)

depend on the particular number x, From (1) we see that we can find the logarithm of any number for the base e by dividing its logarithm for the base by M. The number M is called the modulus of the new system with respect to the original one.

RULE. To find the logarithm of a number for a new base c, divide the common logarithm by the modulus of the system whose base is c.