Table of Contents
What is a logarithm?
Logarithm – Please recall that you have earlier learnt about indices. One of the results we learnt is that if 2^{x} = 2^{3}, X=3 and if 4^{n} = 4^{y} then x = y, i.e., if two powers of the same base are equal and the base not equal to 1, 0 or 1, then the indices are equal.
But when 3^{3} = 5^{2,} just by using the knowledge indices we cannot find the numerical value of x. The necessity of the concept of logarithms arises here. Logarithms are useful in long calculations involving multiplication And division.
Definition of logarithm
The logarithm of any positive number to a given base (a positive number not equal to 1) is the index of the power of the base which is equal to that number. If N and a (≠1) are any two positive real numbers and x is some real number, then a^{x} = N. Where x is said to be logarithm of N to the base a. i.e., if = if a^{x} = n , then x = log_{a}N.
From the definition of logs, we get the following results:
When a > 0, b > 0 and b ≠ 1,
 log_{a}a^{n} = n, e.g., log_{4}4^{3}= 3
 _{a}log_{ab} = b, e.g., 2^{log2∧16}
If in a particular relation, all the log expressions are to the same base, we normally do not specify the base.
Systems of logarithm
There are two systems of logarithms, which are used most often.
 Natural logarithms were discovered by Napier. They are calculated to the base e which is approximately equal to 2.71 These are used in higher mathematics.
 Logarithms to the base 10 are known as common logarithms. This systen was introduced by Briggs, a contemporary of Napier
for the rest of this post , we shall use the short form log rather than logarithm.
Properties
 Logarithms are defined only for positive real numbers.
 Logarithms are defined only for positive bases different from 1.
 In log_{ba}_{ ,} neither a nor is negative hut the value of log_{ba} can be negative.
 Loge of the same number to different numbers to be the same base Are different, i.e., if a≠b, then log_{ma}≠log_{mb.} In other words, if log_{ma =} log_{mb, }Then a = b
 Logs of the same number to different bases different values, i.e., if m≠n, then log_{ma} ≠ log_{na} . In other words, if log_{ma} = log_{na,} then m = n.
 Log of 1 to any base is 0.
 Log of a number to the same base is 1
 Log of 0 is not defined.
Laws
 log_{m }(ab) = log_{ma} +log_{mb}
 log_{m (a/b)} = log_{ma} – log_{mb}
 log a^{m} = mlog a
 log_{ba} log_{cb} = log_{ca }(chain rule)
 log_{ba} = log_{ca }/ log_{cb}
Variation of log_{a}^{x} with x
For 1 < a and 0 < p < q, log_{ap} < log_{ap}
For 0 < a < 1 and 0 < p < q, log_{ap }> log_{aq}
Sign of log_{ax} for Different Values of x and a
Strong bases (a > 1)
 If x > 1, log a^{x} is positive.
 If 0 < X < 1, then log a^{x} is negative.
Weak bases (0 < a < 1)
 If a > 1, then log a^{x} is negative.
 If 0 < X < 1, then log_{ax} is positive.
To Find the log of a Number to Base 10
Consider the following numbers:
2, 20, 200, 0.2 and 0.02.
We see that 20 = 10 (2) and 200 = 100 (2)
∴ log 20 = 1 + log2 and log200 = 2 + log2 similarly, log0.2 = 1 + log 2 and log0.02 = 2 + log2
From the tables, we see that log 2 = 0.3010. (Using the tables, this is explained in more detail in later examples.)
∴ log 20 = 1.3010, log 200 = 2.3010, log 0.2 = 1 +0.3010 and log 0.02 = 2 + 0.3010.
We note one points:

Antilog
As log_{28 = }3, 8 is the antilogarithm of 3 to the base 2. Antilog b to base m is m^{b.}
EXAMPLE
Find the antilog of 1.301.
SOLUTION
Step 1: In the antilog table, we find the number 30 in the left hand column. In that row in the column under 1, we find 2000. Step 2: As the characteristic is 1, we place the decimal after two digits from the left. That is, antilog 1.301 = 20.00 If the characteristic was 2, we would place the decimal after three digits from the left. That is, antilog 2.301 = 200.0 If the characteristic was 3, we would place the decimal after four digits from the left. That is, antilog 3.301 = 2000.