MATHEMATICS
CONTENTS LOGARITHM
KEY CONCEPT ................................. ....................................................... ................................. ........... Page – 2 EXERCISE EXERCIS E – I ............................................. ................................................................... ......................... ... Page – 3 EXERCISE – II ............................................ .................................................................. ......................... ... Page – 4 EXERCISE – III .......................................... ................................................................ ......................... ... Page – 5
6 ANSWER KEY ........................................... ................................................................. ......................... ... Page – 6
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Email:
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KEY CONCEPTS (LOGARITHM) THINGS TO REMEMBER : 1.
LOGARITHM OF A NUMBER :
The logarithm of the number N to the base 'a' is the exponent indicating the power to which the base 'a' must be raised to obtain the number N. T h i s n u m b e r i s d e s i g n a t e d a s lo g N. a logaN = x ! ax = N , a > 0 , a " 1 & N > 0
Hence :
REMEMBER
log 102 = 0.3010 log 103 = 0.4771 ln
2 = 0.693
ln 10 = 2.303 If a = 10 , then we write log b rather than log10 b . If a = e , we write ln b rather than loge b . The existence and uniqueness of the number log a N follows from the properties of an exponential functions.
From the definition of the logarithm of the number N to the base 'a' , we have an identity :
a
log a N
=N , a>0 , a"1 & N>0
This is known as the FUNDAMENTAL LOGARITHMIC IDENTITY . NOTE : loga1 = 0 (a > 0 , a " 1) loga a = 1 (a > 0 , a " 1) and log1/a a = - 1 (a > 0 , a " 1) 2.
THE PRINCIPAL PROPERTIES OF LOGARITHMS: Let M & N are arbitrary posiitive numbers , a > 0 , a " 1 , b > 0 , b " 1 and $ is any real number then ; (i) (ii) loga (M . N) = loga M + loga N loga (M/N) = loga M % loga N (iii)
loga M$ =
$ . loga M
NOTE : ! logba . logab = 1 !
*
3. (i)
!
(iv)
logb M = !
logba = 1/logab.
!
logy x . logz y . loga z = logax.
log a M log a b
logba . logcb . logac = 1 x
e ln a = ax
PROPERTIES OF MONOTONOCITY OF LOGARITHM : For a > 1 the inequality 0 < x < y & log a x < loga y are equivalent.
(ii)
For 0 < a < 1 the inequality 0 < x < y & log a x > loga y are equivalent.
(iii)
If a > 1 then loga x < p
(iv)
If a > 1 then log ax > p
(v)
If 0 < a < 1 then loga x < p
(vi)
If 0 < a < 1 then log ax > p
& & & &
0 < x < ap x > ap x > ap 0 < x < ap
NOTE THAT : ! If the number & the base are on one side of the unity , then the logarithm is positive ; If the number & the base are on different sides of unity, then the logarithm is negative. !
The base of the logarithm ‘a’ must not equal unity otherwise numbers not equal to unity will not have a logarithm & any number will be the logarithm of unity.
!
For a non negative number 'a' & n ' 2 , n ( N
*
Will be covered in detail in QUADRATIC EQAUTION
n
a = a1/ n.
Logarithm
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EXERCISE – I
[STRAIGHT OBJECTIVE TYPE Q.1
1 log
bc
abc
+ log
1 ca
log2 3
If 5 x (A) 2
Q.3
If N + 2 (A) 20
(C) 2
(D) 4
+ 3log2 x = 162 then logarithm of x to the base 4 has the value equal to (C) % 1 (B) 1 (D) 3/2 log 70 9800
log 70 140
5
7
log70 2
, then N is equal to (C) 18
(D) 40
% 1 , ln 3x % 3 ln 2 x % 5 ln x , 7 + 0 (B) 1
(C) 2
(D) 3
2 5 / Number of cyphers after decimal before a significant figure starts in 0 1 4 . [Use: log102 = 0.3010] (A) 6 (B) 7
(C) 8
%100
is equal to
(D) 9
The sum of all values of x satisfying the equation 3
log0.03 )log 2 x * (A) 16 Q.7
has the value equal to
(B) 60
log 2 )ln x *
(A) 0
Q.6
abc
The number of value(s) of x satisfying the equation
4
Q.5
ab
(B) 1
Q.2
Q.4
+ log
abc
(A) 1/2
1
% 4)log 2 x * 2 , 3)log 2 x * , 1 = 0, is
(B) 15
(C) 10 log3 x
Number of values of x satisfying the equation 5 · 3 (A) 0
(B) 1
(D) 11
% 21%log2 x = 3, is
(C) 2
(D) 3
[COMPREHENSION TYPE] Paragraph for question nos. 8 to 10 logMN = $ + 3, where $ is an integer & 3 ( [0, 1) Q.8
If M & $ are prime & $ + M = 7 then the greatest integral value of N is (A) 64 (B) 63 (C) 125 (D) 124
Q.9
If M & $ are twin prime & $ + M = 8 then the greatest integral value of N is (A) 624 (B) 625 (C) 728 (D) 729
Q.10
If M & $ are relative prime & $ + M = 7 then minimum integral value of N is (A) 25 (B) 32 (C) 6 (D) 81
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Paragraph for question nos. 11 to 13 A denotes the product xyz where x, y and z satisfy log3x = log5 – log7 log5y = log7 – log3 log7z = log3 – log5 B denotes the sum of square of solution of the equation log2 (log2x6 – 3) – log2 (log2x4 – 5) = log23 C denotes characterstic of logarithm log2 (log23) – log2 (log43) + log2 (log45) – log2 (log65) + log2 (log67) – log2(log87)
Q.11
Q.12
Q.13
The value of A + B + C is equal to (A) 18 (B) 34
(C) 32
(D) 24
The value of log2A + log2B + log2C is equal to (A) 5 (B) 6 (C) 7
(D) 4
The value of | A – B + C | is equal to (A) – 30 (B) 32
(D) 30
(C) 28
[MULTIPLE OBJECTIVE TYPE] Q.14
2 1
2 2 / /- - = 2, is x 1 . .
The value of x satisfying the equation )log 2 2x * 00 log 2 x , log 2 0 (A) a prime number (C) an even number
2
(B) a composite number (D) an odd number
[INTEGER TYPE / SUBJECTIVE TYPE] Q.15 (a)
)
log (1331)
Let A = log11 11 D = 10
log100 (16 )
11
*,
B = log385(5) + log385(7) + log385(11), C = log 4 )log 2 (log 5 625) * , AD
(b)
. BC Let log7(5a) – log7(a – 4) = 1 and e2b + 5eb = a. If b = ln k where k
Q.16
For 0 < a " 1, find the number of ordered pair (x, y) satisfying the equation log loga y
Q.17
% loga
. Find the value of
x
( N, find
k.
a
2
x,y =
1 2
and
+ log a 2 4 .
Let N be the number of integers whose logarithms to the base 10 have the characteristic 5, and M the number of integers the logarithms to the base 10 of whose reciprocals have the characteristic – 4. Find (log10 N – log10M).
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[MATCH THE COLUMN] Column-I
Q.18
Column-II
(A)
The expression x + log 2 log9 6 , 6 , 6 , ..... 4 simplifies to
(B)
The number N = 2
(C)
The expression
(D)
The number N =
(A)
Column-I For positive real numbers a (a > 1), let pa and qa be the maximum and minimum values, respectively, of
Q.19
log 3 · log 4 · log 5......... · log99 100 2
1 log 5 3
3
,
4
1 log 6 3
%
1 log10 3
(B)
6+
2 1
log 2 01 ,
n 1
2 0 30 1
The value of expression
Q.20 (A) (B)
a prime
(R)
a natural
(S)
a composite Column-II (P) 3
, then the value of a is
(Q)
4
(R)
6
is equal to (S)
8
- is equal to
(D)
%2
2
(Q)
n .
If A = log
log 27 8
1
an integer
1 /
(C)
3
simplifies to
2 , 5 % 6 % 3 5 , 14 % 6 5 simplifies to
loga(x) for a 5 x 5 2a. If pa – qa = 1023
simplifies to
(P)
3 3 3 3
log32 243
/ - , then the value of .
log
2
)8A , 1*
(T)
%5
log625 81
,3
log9 25
+
2
log 2 9
, 3log 4 25 % 5log 4 9 , is
Column-I The value(s) of x, which does not satisfy the equation 2 lo g (x2 – x) – 4 log2(x – 1) log2x = 1, is (are) 2 The value of x satisfying the equation
2
log 2 e
ln
5
log5 7
log 7 10
10
Column-II (P) 2
(Q)
3
(R)
4
(S)
5
(T)
6
log10 (8 x % 3)
= 13, is
(C)
2 1 1 / 0 - is less than , The number N = 0 1 log 2 7 log 6 7 .
(D)
Let l = (log34 + log29)2 – (log34 – log29)2 and m = (0.8) 1 , 9log3 8
)
*
log65 5
then (l + m) is divisible by
Logarithm
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EXERCISE – II
Q.1
A denotes the value of
Let
2 ab , log10 0 0 1
% 4(a , b) /2 2
(ab)
+
2 ab % log10 0 0 1
% 4(a , b) /2 .
(ab)
2
when a = 43 and b = 57 B denotes the value of the expression 2 log6 18 · 3log6 3 .
and
Find the value of (A · B).
2 6 / - in terms of x and y.. 1 5 .
Q.2(a) If x = log34 and y = log 53, find the value of log310 and log3 0
log2 5
(b) If k
Q.3
Solve (a) (b) (c)
(log2 5)
= 16, find the value of k
2
.
for x: log10 (x2 % 12x + 36) = 2 91+logx % 31+logx % 210 = 0 ; where base of log is 3. log10 (2x + x – 41) = x (1 – log105).
Q.4
Simplify: (a) log1/3 4 729.3 9 %1.27 %4 / 3 ;
Q.5
Solve for x:
(b) a
(b) If loge log5 [ 2 x % 2 , 3] = 0
(a) If log4 log3 log2 x = 0 ;
Q.6
Find the value of the expression
Q.7
Simplifythe following: (a) 4
5log
(c)
Q.8
4
5
)3% 6 *%6log8 ) 2
)2 *
log1 / 5 1
,log
3% 2
2 log 4 ( 2000)
6
3 log5 (2000)
7, 3
(b)
,log1 / 2
1
.
10, 2 21
If a, b, c are positive real numbers such that a value of
,
*
4 2
log b logb N logb a
log 3 7
6
81
1 log 9 5
3
,3
log
6
3
409
(d) 49
= 27 ; b
.
1%log 7 2
log 7 11
+ 5
2 ) 1
.0
7
2 log 25 7
*
%)125*log 25 6 / .
% log 5 4
= 49 and c
log11 25
= 11 . Find the
2 a (log 3 7 )2 , b (log7 11) 2 , c (log11 25)2 / . 0 1 .
Q.9
5logx – 3logx-1 = 3logx+1 – 5logx-1, where the base of logarithm is 10.
Q.10
Let y =
log 2 3 · log 2 12 · log 2 48 · log 2 192 , 16
log212 · log248 + 10. Find y ( N.
–
Logarithm
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Q.11
(a)
If 'x' and 'y' are real numbers such that, 2 log(2y – 3x) = log x + log y, find
(b)
If a = log1218 & b = log 2454 then find the value of ab + 5 (a % b). (log9 x )2 %
Q.12
Find the sum of all solutions of the equation 3
Q.13
(a) (b)
Q.14
Q.1
2
log x ,5 9
+3
y
.
3.
(c)
Given : log1034.56 = 1.5386, find log103.456 ; log100.3456 & log100.003456. Find the number of positive integers which have the characteristic 3, when the base of the logarithm is 7. If log102 = 0.3010 & log103 = 0.4771, find the value of log10(2.25).
(d)
If N = antilog3 log6 antilog
5
(log5 1296) , then find the characteristic of logN to the base 2.
If log102 = 0.3010, log103 = 0.4771. Find the number of integers in : (a) 5200
Q.15
9
x
(b) 615
&
( c ) th e n u m b e r o f z e r o s a f t e r t h e d e c i m a l in 3
%100.
If (x1, y1) and (x2, y2) are the solution of the system of equation log225(x) + log64(y) = 4 logx(225) – logy(64) = 1, then find the value of log30(x1y1x2y2). EXERCISE – III Let (x0, y0) be the solution of the following equations
( 2 x )l
n2
+ (3y)ln 3
3ln x = 2 ln y. Then x0 is (A)
1 6
(B)
1 3
(C)
1 2
(D) 6 [JEE 2011, 3]
Logarithm
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ANSWER
SHEET
EXERCISE – I Q.1
B
Q.2
D
Q.3
D
Q.4
C
Q.5
D
Q.6
D
Q.7
B
Q.8
D
Q.9
C
Q.10
C
Q.11
B
Q.12
A
Q.13
D
Q.14
AC
Q.16
0002
Q.17
0002 Q.19
(A) Q; (B) T ; (C) S; (D) P
Q.15(a) 0024 ; (b) 0002 Q.18
(A) P, (B) P, R, S, (C) P, R, (D) P, Q, R
Q.20
(A) Q, R, S, T; (B) P; (C) Q, R, S, T; (D) P, R, S
EXERCISE – II Q.1
xy , 2
12
Q .4 (a)
%1
Q .2 (a)
2y
25
Q.11(a) 4/9 ; (b) 1 (a) 140
2
Q.12
(b) 12
2196
(c) 47
Q.3 (a) x = 16 or x = % 4 (b) x = 5 (c) x = 41
; (b) 625
(b) logbN Q.5 (a) 8 (b) x = 3
Q.7 (a) 9, (b) 1, (c) 6, (d)
Q.14
,
2y
xy , 2 y % 2
Q.8
Q.6 1/6 469
Q.9
x = 100
Q.10
y=6
Q.13 (a) 0.5386; 1.5386 ; 3.5386 (b) 2058 (c) 0.3522 (d) 3 Q.15
12
EXERCISE – III Q.1
C
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Logarithm
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