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01-LOGARITHM, SURDS-(E)-theory

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 1.1.1 Definition.

“The Logarithm of a given number to a given base is the index of the power to which the base must be raised in order to equal the given number.”

If and  then logarithm of a positive number N is defined as the index x of that power of ‘a‘ which equals N i.e., and

It is also known as fundamental logarithmic identity.

The function defined by is called logarithmic function.

Its domain is  and range is R. a is called the base of the logarithmic function.

When base is ‘e‘ then the logarithmic function is called natural or Napierian logarithmic function and when base is 10, then it is called common logarithmic function.

Note : q The logarithm of a number is unique i.e. No number can have two different log to a given base.

q or

 1.1.2 Characteristic and Mantissa.

(1) The integral part of a logarithm is called the characteristic and the fractional part is called mantissa.


(2) The mantissa part of log of a number is always kept positive.

(3) If the characteristics of  be n, then the number of digits in N is (n+1)

(4) If the characteristics of  be (– n) then there exists (n – 1) number of zeros after decimal part of N.


Example: 1        For to be defined ‘a‘ must be                                                                         [IIT 1990]                        [MP PET 1995]

(a)   Any positive real number                                       (b)   Any number

(c)                                                                   (d)   Any positive real number

Solution: (d)       It is obvious (Definition).

Example: 2       Logarithm of to the base is

(a)   3.6                              (b)   5                             (c)   5.6                              (d)   None of these

Solution: (a)        Let x be the required logarithm , then by definition


                   Here, by equating the indices,   ,  

 1.1.3 Properties of Logarithms.

Let m and n be arbitrary positive numbers such that   then

(1)                                   (2)

(3) or       (4)

(5)                             (6)

(7)                                                      (8)

(9)                                          (10)  ,

(11) ,      and


Example: 3       The number is                                                                                                                     [IIT 1990]


(a)   An integer                     (b)   A rational number         (c)   An irrational number        (d)   A prime number


Solution: (c)        Suppose, if possible, is rational, say where p and q are integers, prime to each other.

Then, ,

Which is false since L.H.S is even and R.H.S is odd. Obviously  is not an integer and hence not a prime number

Example: 4       If  then is equal to                                                                                       [Roorkee 1999]

(a)                       (b)                       (c)                          (d)

Solution: (b)       = = =

Example: 5       If  , then relation between a and b will be                                        [UPSEAT  2000]

(a)                            (b)                         (c)                           (d)

Solution: (a)


Example: 6       If , the number of digits in is                                                                                [IIT 1992]

(a)   18                               (b)   19                            (c)   20                              (d)   21

Solution: (c)        Let is

Taking log both the sides,

\ Number of digits in

Example: 7       Which is the correct order for a given number in increasing order                                        [Roorkee 2000]


(a)                                   (b)

(c)                                   (d)

Solution: (b)       Since 10, 3, e, 2 are in decreasing order

Obviously, are in increasing order.

 1.1.4 Logarithmic Inequalities.

(1) If                                     (2) If

(3) If                           (4) If

(5) If                                 (6) If

(7) If                                (8) If


(10)    if base p is positive and >1 or if base p is positive and < 1 i.e.,  

In other words, if base is greater than 1 then inequality remains same and if base is positive but less than 1 then the sign of inequality is reversed.


Example: 8       If which one of the following is correct                                   [W.B. JEE 1993]

(a)                            (b)                         (c)                            (d)   None of these

Solution: (c)



Clearly  , \

Example: 9       If then x lies in the interval

(a)                            (b)   (– 2, –1)                     (c)   (1,  2)                           (d)   None of these

Solution: (a)

or  or or

As base is less than 1, therefore the inequality is reversed, now x>2 x lies in .




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