Logarithm – Please recall that you have earlier learnt about indices. One of the results we learnt is that if 2x = 23, X=3 and if 4n = 4y 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 33 = 52, 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 ax = N. Where x is said to be logarithm of N to the base a. i.e., if = if ax = n , then x = logaN.
From the definition of logs, we get the following results:
When a > 0, b > 0 and b ≠ 1,
- logaan = n, e.g., log443= 3
- alogab = b, e.g., 2log2∧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.
- Logarithms are defined only for positive real numbers.
- Logarithms are defined only for positive bases different from 1.
- In logba , neither a nor is negative hut the value of logba can be negative.
- Loge of the same number to different numbers to be the same base Are different, i.e., if a≠b, then logma≠logmb. In other words, if logma = logmb, Then a = b
- Logs of the same number to different bases different values, i.e., if m≠n, then logma ≠ logna . In other words, if logma = logna, 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.
- logm (ab) = logma +logmb
- logm (a/b) = logma – logmb
- log am = mlog a
- logba logcb = logca (chain rule)
- logba = logca / logcb
Variation of logax with x
For 1 < a and 0 < p < q, logap < logap
For 0 < a < 1 and 0 < p < q, logap > logaq
Sign of logax for Different Values of x and a
Strong bases (a > 1)
- If x > 1, log ax is positive.
- If 0 < X < 1, then log ax is negative.
Weak bases (0 < a < 1)
- If a > 1, then log ax is negative.
- If 0 < X < 1, then logax 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:
As log28 = 3, 8 is the antilogarithm of 3 to the base 2. Antilog b to base m is mb.
Find the antilog of 1.301.
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.