Logarithms Notes for Competitive Exams Preparation

Logarithms Notes for Competitive Exams Preparation

Logarithms:

In the Equation a^x = N, we are expressing N in terms of a and x. The same equation can be re-written as a = N ^1/x . Here we are expressing "a" in terms of N and x. But, among a, x and N, by normal algebraic methods known to us, we cannot express x in terms of the other two parameters a and N. This is where logarithms come into the picture. When a^x = N, then we say x= logarithm of N to the base a, and write it as x= log_aN. The definition of logarithm is given as "the logarithm of any number to a given base is the index of the power to which the base must be raised on order in order to equal the given number".

Thus, if a^x = N then x = log _a N(log N to the base a). This is the basic definition of a logarithm.

This basic definition of logarithm is very useful in solving a number of problems on logarithms.

Example of a logarithm : 216 = 6³ can be expressed as log ₆ 216 = 3.

Since logarithm of a number is a value, it will have an "integral" part and a "decimal" part. The integral part of the logarithm of a number is called the Characteristic and the decimal part of the logarithm is called the Mantissa.

Logarithms are defined only for positive number. There are no logarithms defined for ZERO or Negative Numbers.

Logarithms can be expressed to any base. Logarithms from one base can be converted to logarithms to any other base. (One of the formulae given below will help do this conversion). However, there are two types of logarithms that are commonly used.

i) Natural Logarithms or Napierian Logarithms:
These are logarithms expressed to the base of a number called "e"

ii) Common Logarithms:
These are logarithms expressed to the base 10. For most of the problems under Logarithms, it is common logarithms that we deal with. In examinations also, if logarithms are given without mentioning any base, it can normally be taken to be logarithms to the base 10.

 Given below are some Important Basic Rules/Formulae in logarithms:

1. log _a a=1 (logarithm of any number to the same base is 1)
2. log _a1=0 (log of 1 to any base is 0)
3. log _a( mn ) = log_a m +log_a n
4. log _a(m/n) = log _am-log_an
5. log _am ^p = p log _a m
6. log _a ^q m ^p = p/q log _a m
7. log _a m = log _bm / log _ba
8. log _a b = 1 / log _ba
9. a ^log _a N = N

Worked out Examples:

1) If log (x-3)+log(x+2)= log 6 , then find the value of x.

Sol:-
           log(x-3)(x+2) = log 6

(x-3)(x+2) = 6
x²-x-12=0
(x-4) (x-3) = 0
x = 4 or x = -3
If x = -3, log(x+2) does not exist, as logarithms are not defined for negative values.

Therefore x = 4.

2) Prove that a=b, if 2 log 2+log a+log b=2 log (a+b).

sol:-
         log (2² x a  x b)=log (a+b)²
4ab = a² + b² +2ab
a² + b² - 2ab = 0
(a-b)² = 0
a-b = 0
Therefore a = b.

3) If log (3x - 5) = log (2x+3) find x.

Sol:- 
         log(3x-5) = log (2x+3)
3x-5 = 2x+3
x= 8.

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