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L O GA R I T H M

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Topic
Page No.
LOGARITHM
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Basic Mathematics
Historical Development of Number System
Logarithm
Principal Properties of Logarithm
Basic Changing theorem
Logarithmic equations
Common & Natural Logarithm
Characteristic Mantissa
Absolute value Function
Solved examples
Exercise
Answer Key
13.
Hints & Solutions
L O GA R I T H M
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ACC-MT- LOGARITHM
LOGARITHM
BASIC MATHEMATICS :
Remainder Theorem :
Let p(x) be any polynomial of degree geater than or equal to one and 'a' be any real number.
If p(x) is divided by (x – a), then the remainder is equal to p(a).
Factor Theorem :
Let p(x) be a polynomial of degree greater than or equal to 1 and 'a' be a real number such that
p(a) = 0, then (x – a) is a factor of p(x). Conversely, if (x – a) is a factor of p(x), then p(a) = 0.
Note : Let p(x) be any polynomial of degree greater than or equal to one. If leading coefficient of p(x) is 1
then p(x) is called monic. (Leading coefficient means coefficient of highest power.)
SOME IMPORTANT IDENTITIES :
(1)
(a + b) 2 = a 2 + 2ab + b2 = (a – b)2 + 4ab
(2)
(a – b)2 = a2 – 2ab + b2 = (a + b)2 – 4ab
(3)
a2 – b2 = (a + b) (a – b)
(4)
(a + b)3 = a3 + b3 + 3ab (a + b)
(5)
(a – b)3 = a3 – b3 – 3ab (a – b)
(6)
a3 + b3 = (a + b)3 – 3ab (a + b) = (a + b) (a2 + b2 – ab)
(7)
a3 – b3 = (a – b)3 + 3ab (a – b) = (a – b) (a2 + b2 + ab)
(8)
1 1 1
(a + b + c)2 = a2 + b2 + c2 + 2 (ab + bc + ca) = a2 + b2 + c2 + 2abc .
a b c
(9)
a2 + b2 + c2 – ab – bc – ca =
(10)
a3 + b3 + c3 – 3abc = (a + b + c) (a2 + b2 + c2 – ab – bc – ca)
=
1
2
2
2
2 ( a b ) ( b c) ( c a )
1
(a + b + c) (a b) 2 (b c) 2 (c a ) 2
2
If (a + b + c) = 0, then a3 + b3 + c2 = 3abc.
(11)
a4 – b4 = (a2 + b2) (a2 – b2) = (a2 + b2) (a – b) (a + b)
(12)
If a, b 0 then (a – b) =
(13)
a4 + a2 + 1 = (a4 + 2a2 + 1) – a2 = (a2 + 1)2 – a2 = (a2 + a + 1) (a2 – a + 1)
a b
a b

2
ACC-MT- LOGARITHM
Definition of Indices :
The product of m factors each equal to a is represented by am . So, am = a · a · a ........ a ( m times).
Here a is called the base and m is the index (or power or exponent).
Law of Indices :
(1)
am + n = am · an, where m and n are rational numbers.
(2)
a–m =
(3)
1
, provided a 0.
am
a0 = 1, provided a 0.
(4)
am – n =
(5)
am
, where m and n are rational numbers, a 0.
an
(am)n = amn.
p
q
(6)
a ap
(7)
(ab)n = an bn.
q
Intervals :
Intervals are basically subsets of R (the set of all real numbers) and are commonly used in solving
inequaltities. If a , b R such that a < b, then we can defined four types of intervals as follows :
Name
Open interval
Representation
(a, b)
Discription.
{x : a < x < b} i.e., end points are not included.
Close interval
[a, b]
{x : a x b} i.e., end points are also included.
This is possible only when both a and b are finite.
Open-closed interval
(a, b]
{x : a < x b} i.e., a is excluded and b is included.
Closed-open interval
[a, b)
{x : a x < b} i.e., a is included and b is excluded.
Note :
(1)
The infinite intervals are defined as follows :
(i)
(a, ) = {x : x > a }
(ii)
(iii)
( – , b) = {x : x < b}
(iv)
(v)
(– , ) = {x : x R}
[a, ) = {x | x a }
(– , b] = {x : x b}
(2)
x {1, 2} denotes some particular values of x, i.e., x = 1, 2.
(3)
If their is no value of x, then we say x (i.e., null set or void set or empty set).

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