If log M = x, then M is called the antilogarithm of x and is written as M = antilog x.

For example, if log 39.2 = 1.5933, then antilog 1.5933 = 39.2.

If the logarithmic value of a number be given then the number can be determined from

the antilog-table. Antilog-table is similar to log-table.

Only difference is in the extreme left-hand column which ranges from .00 to .99.

Example on antilogarithm:

1. Find antilog 2.5463.

Solution:

Clearly, we are to find the number whose logarithm is 2.5463. For this consider the

mantissa .5463.

Now find .54 in the extreme left-hand column of the antilog-table (see four-figure antilog-

table)

and then move horizontally to the right to the column headed by 6 of the top-most row

and read the number 3516.

Again we move along the same horizontal line further right to the column headed by 3 of

mean difference and read the number 2 there.

This 2 is now added to the previous number 3516 to give 3518. Since the characteristic is 

2, there should be three digits in the integral part of the required number.

Therefore, antilog 2.5463 = 351.8.

2. If log x = -2.0258, find x.

Solution:

In order to find the value of x using antilog-table, the decimal part (i.e., the mantissa)

must be made positive. For this we proceed as follows:

log x = -2.0258 = - 3 + 3 - 2.0258

= - 3 + .9742 =3.9742

Therefore, x = antilog 3.9742.

Now, from antilog table, we get the number corresponding to the mantissa

.9742 as (9419 + 4) = 9423.

Again the characteristic in log x is (- 3).

Hence, there should be two zeroes between the decimal point and the first significant digit

in the value of x.

Therefore, x = .009423.

A logarithm is a power to which a number must be raised in order to get some other

number.

In other words, if a is a positive real number, other than 1 and a^m = x,

then we write: m = log_a x

and we say that the value of log x to the base a is m.

Ex 1. 10³ = 1000 => log₁₀ 1000 = 3

Ex 2. 3⁴ = 81 => log₃ 81 = 3

Ex 3. 2^-3 = 1/8 => log₂ (1/8) = -3

There are some rules in logarithm listed below:

1.  log_a (x * y) = log_a x + log_a y

2.  log_x x = 1

3.  log_a (x/y) = log_a x - log_a y

4.  log_a 1 = 0

5.  log_a (x^p ) = p * log_a x

6.  log_a x = 1/log_x a

7.  log_a x = log_b x/log_b a = log x/log a

It is assumed that when base is not given, it is taken as 10