CENGAGE/GTEWANIMATHSSOLUTIONS

CHAPTERLOGARITHM||TRIGONOMETRY

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1

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_ExponentialFunctions

Solve:

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2

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_ExponentialFunctions

Solve:

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3

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_ExponentialFunctions

Findthesmallestintegralvalueof satisfying

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4

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_ExponentialFunctions

Findthenumberofsolutionsofequation

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5

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Findthevalueof

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|x − 3|3x2− 10x+ 3 = 1

( )x

^ (2 − 2x) < 1/4.12

x (x − 2)x2 − 6x+ 8) > 1

(2x− 3)2x = 1

(log)2√31728.

6

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Provethat

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7

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Arrange indecreasingorder.

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8

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Provethatnumber isanirrationalnumber.

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9

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Which of the following numbers are positive/negative? (ii) (iii)

(v)

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10

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Findthevalueoflog

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

If and ,thenprovethat

< (log)103 < .13

12

(log)25, (log)0 . 55, (log)75, (log)35

(log)27

(log)27 (log)0 .23

(log)1 /3( )15

(log)43 (log)2((log)29)

tan 10log tan 20.......... . log tan 890

(log)a3 = 2 (log)b8 = 3 (log)ab = (log)34.

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12

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

If find intermsof

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13

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

If

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14

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Solve:

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15

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Findthenumberofsolutiontoequation

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16

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Find the number of solutions of the following equations:

a b a 3

(log)3y = xand(log)2z = x, 72x y and z.

= , provethatxyyx

= zxyz = xzzx

x(y + z− x)log x

y(z + x − y)log y

z(x + y − z)log z

2(25)x − 5(10x) + 2(4x) ≥ 0.

(log)2(x+ 5) = 6 − x:

x− (log)0 .5x = 112

x2 − 4x+ 3 − (log)2x = 0

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17

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Findthevalueof

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18

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If then find the relation between

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19

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If arerealnumberssuchthat ,thenfind .

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20

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Whatislogarithmof tothebase

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21

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Whichisgreater:

(log)2(293 − 2) + (log)2(1233 + 4 + 493).

(log)e( ) = ((log)ea + (log)eb),a + b

212

aandb.

xandy 2 log(2y − 3x) = logx + log yx

y

3245 2√2?

x = (log)35 or y = (log)1725?

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22

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

,thenfindxintermsofy.

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23

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If

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24

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If thenfindthevalueof

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25

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Findthevalueof

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26

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If thenprovethat

y = 21

(log )x4

n > 1, thenprovethat + + + = .1

(log)2n

1

(log)3n

1

(log)53n

1

(log)53 !n

ax = b, by = c, cz = a, xyz.

((log)34)((log)45)((log)56)((log)67)((log)78)((log)89).

y2 = xz and ax = by = cz, (log)ab = (log)bc

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27

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Simplify:

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28

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Suppose andarenotequal to1and Find

thevalueof

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29

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If thenfind

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30

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Findthevalueof

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+ +1

1 + (log)abc

1

1 + (log)bca

1

1 + (log)cab

x, y, z = 0 logx + log y + log z = 0.

+ + +1

xlogy

1^ (log z)

1

ylog z

1^ (log x)

1

z logx

1^ (log y)

(log)1227 = a, (log)616∫ermsofa

( )1 + ( log ) 72

+ 5− 1( log ) ( ) ( 7 )1

4915

31

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Findthevalueof

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32

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Provethat

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33

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Provethat:

if

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34

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Whichofthefollowingpairsofexpressionaredefinedforthesamesetofvaluesof?

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35

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve

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81 ( 1/ log5 3) + (27log9 36) + 3( )4

log7 9

02 ( log ) 2 x − 3log _ (27)(x2 + 1)

3− 2x >

14

74 ( log) 49x − x − 1

2√( log) a4√ab+ ( log ) b4√ab− ( log ) a4√ + ( log ) b4√

.

√(log)ab

= {2 if b ≥ a > 1 and 2loga ( b )

ba

a

b

1

x

f1(x) = 2(log)2xandf2(x) = (log)10x2 f1(x) = (log)2

×andf2(x) = 2f1(x) = (log)10(x − 2) + (log)10(x− 3)andf2(x ) = (log)10(x

− 2)(x − 3).

(log)48 + (log)4(x + 3) − (log)4(x − 1) = 2.

36

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve

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37

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve

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38

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve

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39

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve:

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40

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve:

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41

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve: (baseise)

log( − x) = 2 log(x+ 1).

(log)2(3x − 2) = (log) x12

2x+ 227x/ (x− 1 ) = 9

(log)2(4.3x − 6) − (log)2(9x − 6) = 1.

6((log)x2 − (log)4x) + 7 = 0.

4 ( log ) 2 log x = logx − (log x)2

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42

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve:

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43

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve

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44

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve:

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45

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve:

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4(log) (√x) + 2(log)4x(x2) = 3(log)2x(x

3).x

2

4 ( log ) 9x − 6x ( log ) 92 + 2 ( log) 327 = 0

xlos2√x = (2. x ( log ) 2x)14

14

|x − 1| ( log) 10x ^ 2 − (log)10x2 = |x − 1|3

46

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve

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47

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve

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48

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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49

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve

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50

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve

(log)2(x − 1) > 4.

(log)3(x − 2) ≤ 2.

(log)0 .3(x2 − x+ 1) > 0

1 < (log)2(x − 2) ≤ 2.

log2|x − 1| < 1

(log)0 . 2|x − 3| ≥ 0.

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52

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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53

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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54

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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55

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve

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56

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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.

(log)2 > 0x − 1x − 2

(log)0 . 5 < 03 − x

x + 2

(log)3(2x2 + 6x − 5) > 1

(log)0 . 04(x − 1) ≥ (log)0 . 2(x− 1)

(log) (x+ 3) (x2 − x) < 1

57

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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58

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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59

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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60

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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61

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solve:

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

2(log)3x − 4(log)x27 ≤ 5 (x > 1)

(log)x+ ( ) > 01x

log2(x − 1)x + 2

(log) ( log ) 2 ( ) (x2 − 10x+ 22) > 0x

x

(log)0 . 1((log)2( ) < 0x2 + 1x − 1

≤ 1.x − 1

(log)3(9 − 3x) − 3

62 Solve:

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63

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

Write the characteristic of each of the following numbers by using their standardforms:1235.5 (ii) 346.41 (iii) 62.723 (iv) 7.123450.35792 (vi)0.034239(vii)0.002385(viii)0.0009468

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64

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

Writethesignificantdigitsineachofthefollowingnumberstocomputethemantissaoftheirlogarithms:3.239(ii)8(iii)0.9(iv)0.020.0367(vi)89(vii)0.0003(viii)0.00075

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65

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

Findthemantissaofthelogarithmofthenumber5395

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66

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

Findthemantissaofthelogarithmofthenumber0.002359

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67

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

Use the logarithm tables to find the logarithm of the following numbers (1)25795(ii)25.795

( )log

_ (10)a2 + 2 >12

3

2 ( log ) 10 ( −a )

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68

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

Find the antilogarithm of each of the following: 2.7523 (ii) 3.7523 (iii)5.7523(iv)0.75231.7523(vi)2.7523(vii)3.7523

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69

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

Evaluate

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70

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

Usinglogarithms,findthevalueof6.45x981.4

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71

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

Let Find thecharacteristicandmantissaof the logarithmof to thebase10.Assume

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72

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

If then find the number of digits in

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(72.3) if log 0. 723 = 1. 8591.13

x = (0. 15)20. x(log)102 = 0. 301and(log)103 = 0. 477.

(log)102 = 0. 30103, (log)103 = 0. 47712,312 ⋅ 28

73

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

In the 2001 census, the population of India was found to be If thepopulationincreasesattherateof2.5%everyyear,whatwouldbethepopulationin2011?

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74

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

FindthecompoundinterestonRs.12000for10yearsattherateof12%perannumcompoundedannually.

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75

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

If is the number of natural numbers whose logarithms to the base 10 have thecharacteristic is the number of natural numbers logarithms of whosereciprocals to the base 10 have the characteristic , then find the value of

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76

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

Let denoteantilog_320.6andMdenotethenumberofpositiveintegerswhichhavethe characteristic 4, when the base of log is 5, and N denote the value of

Findthevalueof

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77

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If ,provethat

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8. 7 ⋅ 107.

PpandQ

−qlog10 P − (log)10Q.

L

49 ( 1− ( log) 72 ) + 5 − ( log ) 54 ..

LM

N

x = (log)2aa, y = (log)3a2a, z = (log)4a3a 1 + xyz = 2yz

78

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solvetheequationsfor

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79

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If thenfindthevalueof

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80

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If and ,thenprovethat

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81

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Solve

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82

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Let be positive integers such that If

thenfindthevalueof

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

xandy : (3x)log 3 = (4y)log g4, 4log x = 3logy.

a = (log)1218, b = (log)2454, ab + 5(a − b).

y = a1

1 − ( log ) ax z = a1

1 − ( log )ay x = a1

1− ( log ) az

(log)x2(log)2x2 = (log)4x2.

a, b, c, d (log)ab = and(log)cd = .32

54

(a − c) = 9, (b − d).

√

83 Solve (baseis10).

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84

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

If then find the largest possible value of the expression

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85

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve:

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86

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solvetheinequality

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87

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Findthenumberofsolutionsofequation

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88

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve

√log( − x) = log √x2

a ≥ b > 1,

(log)a( ) + (log)b( ).a

b

b

a

3 ( log9 x ) × 2 = 3√3

√(log)2( ) < 12x − 3x − 1

2x + 3x + 4x − 5x = 0

x ( log ) yx = 2andy ( log ) xy = 16

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89

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve

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90

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solvefor: .

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91

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If (where aredifferentpositiverealnumbers thenfindthevalueof

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92

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If

andnotequalto1,thenprovethat

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

(log)2x2 + (log)42x = − 3/2.

x : (2x) ( log ) b2 = (3x) ( log ) b3

(log)ba(log)ca + (log)ab(log)cb + (log)ac(log)bc = 3 a, b, c≠ 1), abc.

= , whereN > 0andN ≠ 1, a, b, c

> 0

(log)aN

(log)cN

(log)aN − (log)bN

(log)bN − (log)cN

b2 = ac

2

93 Given are positive numbers satisfying Findtherangeofvaluesof

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94

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solve:

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95

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

is a rational number (b) an irrational number a primenumber (d) noneofthese

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96

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

The number of lies between two successiveintegerswhosesumisequalto(a)5(b)7(c)9(c)10

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97

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Antilogarithm

Giventhat thenumberofdigits in thenumber is6601(b)6602(c)6603(d)6604

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aandb 4(log10 a)2 + ((log)2b)2 = 1.

aandb.

(log) ( 2x+ 3 ) (6x2 + 23x + 21) + (log) ( 3x+ 7) (4x2 + 12x+ 9)

= 4

(log)418

N = 6 − (6(log)102 + (log)1031)

log(2) = 0. 3010, 20002000

98

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If ,thenthevalueof is0(b)1(c)2(d)4

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99

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Thevalueof (b) (c) (d)noneofthese

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100

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If areconsecutivepositiveintegersand thenthevalueof is (b) (c)2(d)1

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101

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If ,thenbisequalto (2) (c) (d)

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102

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If are such that then the value ofisequalto0(b)1(c)2(d)noneofthese

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(21. 4)a = (0. 00214)b = 100 −1a

1b

log ab − log|b| = log a log|a| − loga

a, b, c (log(1 + ac) = 2K,K log b loga

= = ba + (log)43

a + (log)23

a + (log)83

a + (log)43

12

23

13

32

p > 1andq > 1 log(p + q) = logp + log q,log(p − 1) + log(q − 1)

103

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Thevalueof is2(b)3(c)4(d)1

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104

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If then isequal to (b) (c)

(d)

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105

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If then intermsof isequalto(a)

(b) (c)

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106

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Thereexistsanaturalnumber whichis50timesitsownlogarithmtothebase10,then isdivisibleby

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107

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Thevalueof is3(b)0(c)2(d)1

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+ ((log)62)21 + 2(log)32

(1 + (log)32)2

(log)45 = aand(log)56 = b, (log)321

2a + 11

2b + 1

2ab + 11

2ab − 1

(log)102 = a, (log)103 = b (log)0 .72(9. 6) aandb2a + 3b − 15a + b − 2

5a + b − 13a + 2b − 2

3a + b − 22a + 3b − 1

2a + 5b − 23a + b − 1

NN

−(log)224

(log)962

(log)2192

(log)122

108

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

isequal to:9 (b)16 (c)25(d)noneofthese

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109

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If (a) (b)

(c) (d)

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110

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If thenthevalueof equals

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111

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Thevalueof is0(b)1(c)2(d)noneofthese

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112

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If thenthevalueof is1 (b)

(log)x− 1x(log)x− 2(x − 1)(log)x− 12(x− 11) = 2, x

f(x) = log( ), then1 + x

1 − xf(x1)f(x) = f(x1 + x2)

f(x + 2) − 2f(x + 1) + f(x) = 0 f(x) + f(x + 1) = f(x2 + x)

f(x1) + f(x2) = f( )x1 + x2

1 + x1x2

a4b5 = 1 loga(a5b4)

3 ( log ) 45 − 5 ( log ) 43

2x+ y = 6yand3x− 1 = 2y+ 1, (log 3 − log 2)(x− y)

3

(d)noneofthese

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113

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Thevalueof satisfying theequation 1 (b)3 (c)18(d)54

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114

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If then equals odd integer (b) prime number

compositenumber(d)irrational

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115

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If ,thenwhichofthefollowingis/aretrue? (b)

(d)

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116

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

if thevalueof is9(b)12(c)15(d)30

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(log)23 − (log)32 log( )32

x 3√5 ( log5 ) 5( ( log ) 5 ( log ) 5 log5 ( ) )

= 3x

2

√(log)2x − 0. 5 = (log)2√x, x

= =log xb − c

log yc − a

log za − b

zyz = 1

xaybzc = 1 xb+ cyc+ b = 1 xyz = xaybzc

(log)yx + (log)xy = 2, x2 + y = 12, xy

117

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Miscellaneous

If thenall therootsarepositiverealnumbersall the roots lie in the interval (0,100) all the roots lie in the interval [-1,99] noneofthese

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118

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If

2(b)65/8(c)37/6(d)noneofthese

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119

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If ,then 4(b)3(c)2(d)1

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120

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If then isequalto4(b)3(c)8(d)

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

(x + 1) ( log ) 10 ( x+ 1 ) = 100(x+ 1),

(log)2x+ (log)x2 = = (log)2y + (log)y2andx ≠ y, thex

+ y =

103

(log)10[ ] = x[(log)105 − 1]1

2x + x − 1x =

(log)3{5 + 4(log)3(x− 1)} = 2, x (log)216

( log) 3 ( log )

121 If then isequalto

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122

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Equation

onlyoneprimesolutiontworealsolutionsnorealsolution(d)noneofthese

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123

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

The value of for which the equationhascoincidentrootsis

(b) (d)

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124

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Iftheequation issolvedfor intermsof where thenthesumofthesolutionis (b) (d)

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125

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Thenumberofsolutionof is0(b)1(c)2(d)

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

2x ( log) 43 + 3 ( log ) 4x = 27, x

(log)4(3 − x) + (log)0 .25(3 + x) = (log)4(1 − x)

+ (log)0 . 25(2x + 1)has

b2(log) (bx + 28) = − (log)5(12 − 4x − x2)1

25b = − 12

b = 4 or b = − 12 b = 4 or b = − 12 b = − 4 or b = 12

2x + 4y = 2y + 4x y x x < 0,x(log)2(1 − 2x) x + (log)2(1 − 2x) (log)2(1 − 2x)

x(log)2(2x + 1)

xlog _ x(x + 3)2 = 16 ∞

126 Theproductofrootsoftheequation is1(b)(c)1/3(d)

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127

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Let be a real number. Then the number of roots equationis2(b)infinite(c)0(d)1

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128

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Thenumberofrootsoftheequation is1(b)2(c)3(d)0

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129

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

The number of elements in set of all satisfying the equation

(a)1(b)2(c)3(d)0

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130

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Number of real values of satisfying the equation

,is(a) (b) (c) (d)

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131

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If ,then equals (a) (b)

(c) (d)

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= 33

(log8 x)21/4

a > 1a2 ( log) 2x = 15 + 4x ( log ) 2a

(log)3√xx + (log)3x√x = 0

x

xlog3 x2 + ( log3 x ) 2 − 10 = is

1

x2

x

log2(x2 − x) ⋅ log2( ) + (log2 x)2 = 4x− 1x

0 2 3 7

xy2 = 4and(log)3((log)2x) + (log) ((log) y) = 113

12

x 4 816 64

132

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

If are the roots of the equation with then(b) (d)

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133

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

The number of real values of the parameter for which with real coefficients will have exactly one

solutionis 2 1 4 noneofthese

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134

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

implies

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135

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

If , then (a)

(b) (d) (d)noneofthese

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136

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

If then is equal to(b) (c) (d)

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

x1andx2 e2 ⋅ xln x = x3 x1 > x2,x1 = 2x2 x1 = x2

2 (c)2x1 = x22 x2

1 = x32

k

(log16 x)2 − (log)16x + (log)16k = 0(1) (b) (c) (d)

xlog5 x > 5 x ∈

S = {x ∈ N : 2 + (log)2√x + 1 > 1 − (log) √4 − x2}12

S = {1}

S = Z S = N

S = {x ∈ R : ((log)0 . 60. 216)(log)5(5 − 2x) ≤ 0}, S

(2. 5, ∞) [2, 2.5) (2, 2.5) (0, 2.5)

137

Solutionsetoftheinequality is (b) (c)

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138

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

If thentheleastvalueof is4(b)8(d)16(d)32

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139

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Which of the following is not the solution of

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140

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

The solution set of the inequality is

(b) (c) (d)

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Solution set of the inequality is (b)

>1

2x − 11

1 − 2x− 11, ∞) 0, (log)2( )4

3

( − 1, ∞) (0, (log)2( ) ∪ (1, ∞)43

(log)2x + (log)2y ≥ 6, x + y

(log)x( − ) > 152

1x

(a)( , )(b)(1, 2)(c)( , 1)(d)Nonofthese25

12

25

(log)10(x2 − 16) ≤ (log)10(4x − 11) 4, ∞)

(4, 5) ( , ∞)114

( , 5)114

(log)0 . 8((log)6 ) < 0x2 + x

x + 4( − 4, − 3)

( − 3, 4) ∪ (8, ∞) ( − 3, ∞) ( − 4, − 3) ∪ (8, ∞)

141 (d)

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142

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Whichof thefollowing isnot thesolutionof

is (b) (d)noneofthese

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143

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

The true solution set of inequality isequal to (b)

(d)

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144

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

For the rootsof theequation

aregiven (b) (c) (d)

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145

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

The real solutions of the equation is/are 1 (b) 2 (c)(d)

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( − 3, 4) ∪ (8, ∞) ( − 3, ∞) ( − 4, − 3) ∪ (8, ∞)

(log)3(x2 − 2) < (log)3( |x| − 1)32

(√2, 2) ( − 2, − √2) ( − √2, 2

(log) (x+ 1) (x2 − 4) > 1 2, ∞)

(2, )1 + √212

( , )1 − √212

1 + √212

( , ∞)1 + √212

a > 0, ≠ 1, (log)axa + (log)xa2 + (log)a2a

3 = 0a− 4

3 a− 34 a a− 1

2

2x+ 2. 56 −x = 10x ^ 2−(log)10(250) (log)104 − 3

146

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If then is equal to (b) 1/5 (c) 5 (d)noneofthese

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147

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If satisfytheequation then are(a)relativelyprime(b)twinprime(c)coprime(d)if isdefined,then isnotandviceversa

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148

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Which of the following, when simplified, reduces to unity? (a)

(b) .(c) (d)

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149

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

If forpermissiblevaluesof and thenidentifythestatement(s)whichcanbecorrect.(a)If and aretwoirrationalnumbers,then canberational.(b)Ifis rational and is irrational, then can be rational. (c)If is irrational and isrational,then canberational.(d)if arerational,then canberational.

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

(log)kx.

log5k = (log)x5, k ≠ 1, k > 0, x k

p, q ∈ N x√x = (√x)x, pandq

(log)qp (log)pq

(log)105.

log1020 + ((log)102)2 2 log 2 + log 3log 48 − log 4

−(log)5(log)3√5√9

(log) ( )16 √3

2

6427

(log)ax = b a x,a b x a

b x a bx aandb x

(log) ( − .5) = (log) ( + 1)

150 Theequation has(A)tworealsolutions(B)noprimesolution(C)oneintegralsolution(D)noirrationalsolution

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151

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If

(d)noneofthese

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152

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

The value of independent of (b)

independentof dependenton (d)dependenton

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153

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

The inequality is satisfied by (A) only one value of (B)

(d)

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154

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If thenthevalueofis............

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155

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Thevalueof is............

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(log)x+ 1(x − .5) = (log)x− 0. 5(x+ 1)

(log)105 = aand(log)103 = b, then (A)(log)308 =3(1 − a)b + 1

(B)(log)4015 =a + b

3 − 2a(C)(log)24332 =

1 − a

b

(6a ( log ) eb((log)a2b) is(log)b2a

e ( log) ea ( log) eba

b a b

√x ( log) 2 √x ≥ 2 x

x ∈ (0, ( )]14

(C)x ∈ [4, ∞) x ∈ (1, 2)

(log)ab = 2, (log)bc = 2, and(log)3c = 3 + (log)3a, c/ (ab)

(log10 2)3 + log10 8 log10 5 + (log10 5)3

156

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

If (where [ ] represents

thegreatestintegerfunction)equals..............

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157

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Integral value of which satisfies the equation

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158

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Thedifferenceofrootsoftheequation is.......

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159

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Sum of all integral values of satisfying the inequality

is......

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160

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Theleastintegergreaterthan is................

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(log)4A = (log)6B = (log)9(A + B), then [4( )]B

A

x

= log6 54 + (log)x16 = (log)√2x − (log)36( )is..49

((log)27x3)

2= (log)27x

6

x

(3( ) log( 12 − 3x ) ) − (3log x) > 3252

(log)2(15) ⋅ (log) 2 ⋅ (log)316

16

161

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Thereciprocalof is............

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162

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Sumofintegerssatisfying is......

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163

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Numberofintegerssatisfyingtheinequality is.....

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Number of integers satisfying the inequality

+2

(log)4(2000)6

3

(log)5(2000)6

√(log)2x − 1 − (log)2(x3) + 2 > 012

(log) |x − 3| ≻ 112

≤ 101 1

164 is............

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165

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Thevalueof is...........

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166

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Thevalueof is.........

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167

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

Thevalueof is...........

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168

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

The number of positive integers satisfyingis...........

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

2(log) (x− 1) ≤ −12

13

1

(log)x2 −x8

(log)√4 + 2√2√4− 2√229

5( log ) ( )

+ (log)√2 + (log)15

12 4

√7 + √312

1

10 + 2√21

N = −(log)5250

(log)505

(log)510

(log)12505

x + (log)10(2x + 1) = x(log)105 + (log)106

169

The are positive real numbers such that

then the value of is

............

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170

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solvefor

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171

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Solvethefollowingequationofx:

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172

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

If are in arithmetic progression,

determinethevalueof

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173

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Thesolutionoftheequation is...

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CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

x, y, z

(log)2xz = 3, (log)5yz = 6, and(log)xyz = ,23

( )12z

x: 4x − 3x− = 3x+ − 22x− 112

12

2(log)xa + (log)axa + 3(log)a2xa = 0, a > 0

(log)32, (log)3(2x − 5)and(log)3(2x − )72

x.

(log)7(log)5(√x + 5 + √x = 0

2(log) − (log) (0.01) > 1 10 2

174 Theleastvalueoftheexpression for is(a) (b)(c) (d)

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175

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_Fundamental Laws OfLogarithms

If In are in then (b)

arein (d) arein

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176

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

Let be the solution of the following equations:

The is (b) (c) (d)6

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177

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicEquations

Theequation has at leastone realsolutionexactlythreesolutions exactlyoneirrationalsolution complexroots

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178

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicFunctions

If , then (b) (d)

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179

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_LogarithmicInequalities

Let Aninteger satisfying mustbe lessthan..........

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2(log)10x− (log)x(0.01). x > 1 10 2−0. 01 4

(a + c), ln(a − c), ln(a − 2b + c) A.P ., a, b, c, are ∈ A

.P .

a2, b2, c2are ∈ A.P . a, b, c G

.P . a, b, c H

.P .

(x0, y0) (2x)1n2 = (3y)1n3

31nx = 21ny x016

13

12

x( ) ( log2 x ) 2 + ( log2 x ) − ( ) = √2

34

54 (1)

(2) (3) (4)

3x = 4x− 1 x =2(log)32

2(log)32 − 1

2

2 − (log)23

1

1 − (log)432(log)23

2(log)23 − 1

a = (log)3(log)32. k 1 < 2−k+ 3 ( − a )< 2,

180

CENGAGE_MATHS_TRIGONOMETRY_LOGARITHM_FindingLogarithm

Thevalueof is...............

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6 + (log)⎡⎣ ⋅

⎷(4 − )√4 − .......32

1

3√2

1

3√2

1

3√2