Chapter 8 : Logarithms - Selina Solutions for Class 9 Maths ICSE

Mathematics in ICSE Class 9 is one of the most challenging and trickiest subjects of all. It includes complex topics such as logarithms, expansions, indices and Pythagoras Theorem which are difficult to understand for an average student. TopperLearning provides study materials for ICSE Class 9 Mathematics to make the subject easy and help students to clear all their concepts. Our study materials comprise numerous video lessons, question banks, revision notes and sample papers which help achieve success in the examination.

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Chapter 8 - Logarithms Excercise Ex. 8(B)

Question 1

Express in terms of log 2 and log 3:

(i) log 36 (ii) log 144 (iii) log 4.5

(iv) log - log (v) log log + log

Solution 1

(i)

(ii)

(iii)

(iv)

(v)

Question 2

Express each of the following in a form free from logarithm:

(i) 2 log x - log y = 1

(ii) 2 log x + 3 log y = log a

(iii) a log x - b log y = 2 log 3

Solution 2

(i)

(ii)

(iii) 

Question 3

Evaluate each of the following without using tables:

(i) log 5 + log 8 - 2 log 2

(ii) log108 + log1025 + 2 log103 - log1018

(iii) log 4 + log 125 - log 32

 

Solution 3

(i)  

 

 

(ii)  

 

 

 

 

(iii)  

 

 

Question 4

Prove that:

 

Solution 4

 

 

 

Question 5

Find x, if:

x - log 48 + 3 log 2 = log 125 - log 3.

 

Solution 5

 

 

 

 

Question 6

Express log102 + 1 in the form of log10x.

Solution 6

 

 

Question 7

Solve for x:

(i) log10 (x - 10) = 1

(ii) log (x2 - 21) = 2

(iii) log (x - 2) + log (x + 2) = log 5

(iv) log (x + 5) + log (x - 5)

= 4 log 2 + 2 log 3

Solution 7

(i) 

(ii)

(iii)

(iv)

Question 8

Solve for x:

(i)

 

 

(ii)

 

 

(iii)

 

 

(iv)

 

Solution 8

(i)

 

 

 

(ii)

 

 

 

(iii)

 

 

 

(iv)

 

Question 9

Given log x = m + n and log y = m - n, express the value oflog in terms of m and n.

 

Solution 9

 

 

 

 

 

Question 10

State, true or false:

(i) log 1 log 1000 = 0

(ii)

 

 

(iii) If then x = 2

 

 

(iv) log x log y = log x + log y

 

Solution 10

(i)

 

 

 

(ii)

 

 

(iii)

 

 

(iv)

 

Question 11

If log102 = a and log103 = b; express each of the following in terms of 'a' and 'b':

(i) log 12(ii) log 2.25(iii) log

 

(iv) log 5.4(v) log 60(iv) log

 

Solution 11

(i)

 

 

(ii)

 

 

(iii)

 

 

 

(iv)

 

 

(v)

 

 

 

(vi)

 

 

Question 12

If log 2 = 0.3010 and log 3 = 0.4771; find the value of:

(i) log 12(ii) log 1.2(iii) log 3.6

(iv) log 15(v) log 25(vi) log 8

 

Solution 12

(i)

 

 

(ii)

 

 

 

(iii)

 

 

 

(iv)

log space 15 equals log open parentheses 15 over 10 cross times 10 close parentheses
space space space space space space space space space space space equals log open parentheses 15 over 10 close parentheses plus log space 10 space
space space space space space space space space space space space equals log open parentheses 3 over 2 close parentheses plus 1 space space space space space space space space space space space space space space space space space space space space left square bracket because log space 10 equals 1 right square bracket
space space space space space space space space space space space equals log space 3 minus log space 2 plus 1 space space space space space space space space space space space space space left square bracket because log space straight m minus log space straight n equals log open parentheses straight m over straight n close parentheses right square bracket
space space space space space space space space space space space equals 0.4771 minus 0.3010 plus 1
space space space space space space space space space space space equals 1.1761

 

 

(v)

 

(vi)

 

 

 

Question 13

Given 2 log10 x + 1 = log10 250, find :

(i) x(ii) log10 2x

Solution 13

(i)

 

 

 

(ii)

 

 

Question 14

Solution 14

  

Question 15

Solution 15

  

Question 16

Solution 16

  

Chapter 8 - Logarithms Excercise Ex. 8(C)

Question 1

If log10 8 = 0.90; find the value of:

(i) log10 4(ii) log

 

(iii) log 0.125

Solution 1

 

(i)

 

 

(ii)

 

 

(iii)

 

 

Question 2

If log 27 = 1.431, find the value of :

(i) log 9(ii) log 300

Solution 2

 

 

(i)

 

 

(ii)

 

Question 3

If log10 a = b, find 103b - 2 in terms of a.

Solution 3

Question 4

If log5 x = y, find 52y+ 3 in terms of x.

Solution 4

Question 5

Given: log3 m = x and log3 n = y.

(i) Express 32x - 3 in terms of m.

(ii) Write down 31 - 2y + 3x in terms of m and n.

(iii) If 2 log3 A = 5x - 3y; find A in terms of m and n.

 

Solution 5

 

(i)

 

 

 

(ii)

 

 

 

(iii)

Consider space the space given space expression :
2 space log subscript 3 straight A equals 5 straight x space minus space 3 straight y
rightwards double arrow 2 space log subscript 3 straight A equals 5 space log subscript 3 straight m minus 3 log subscript 3 straight n
rightwards double arrow log subscript 3 straight A squared equals log subscript 3 straight m to the power of 5 minus log subscript 3 straight n cubed
rightwards double arrow log subscript 3 straight A squared equals log subscript 3 open parentheses straight m to the power of 5 over straight n cubed close parentheses
rightwards double arrow straight A squared equals open parentheses straight m to the power of 5 over straight n cubed close parentheses
rightwards double arrow straight A equals square root of open parentheses straight m to the power of 5 over straight n cubed close parentheses end root

 

Question 6

Simplify:

(i) log (a)3 - log a

(ii) log (a)3 log a

Solution 6

(i)

(ii)

Question 7

If log (a + b) = log a + log b, find a in terms of b.

Solution 7

Question 8

Prove that:

(i) (log a)2 - (log b)2 = log . Log (ab)

(ii) If a log b + b log a - 1 = 0, then ba. ab = 10

 

Solution 8

(i)

 

 

(ii)

 

 

Question 9

(i) If log (a + 1) = log (4a - 3) - log 3; find a.

(ii) If 2 log y - log x - 3 = 0, express x in terms of y.

(iii) Prove that: log10 125 = 3(1 - log102).

 

Solution 9

(i)

 

 

 

(ii)

 

 

 

(iii)

 

 

Question 10

Solution 10

Given space log space straight x space equals space 2 straight m minus straight n comma space log space straight y equals straight n minus 2 straight m space and space log space straight z equals 3 straight m minus 2 straight n
log fraction numerator straight x squared straight y cubed over denominator straight z to the power of 4 end fraction equals logx squared straight y cubed minus logz to the power of 4
equals log space straight x squared plus log space straight y cubed minus log space straight z to the power of 4
equals 2 log space straight x plus 3 log space straight y minus 4 log space straight z
equals 2 left parenthesis 2 straight m minus straight n right parenthesis plus 3 left parenthesis straight n minus 2 straight m right parenthesis minus 4 left parenthesis 3 straight m minus 2 straight n right parenthesis
equals 4 straight m minus 2 straight n plus 3 straight n minus 6 straight m minus 12 straight m plus 8 straight n
equals negative 14 straight m plus 9 straight n

Question 11

Solution 11

  

Chapter 8 - Logarithms Excercise Ex. 8(D)

Question 1

If log a + log b - 1 = 0, find the value of a9.b4.

 

Solution 1

 

 

Question 2

If x = 1 + log 2 - log 5, y = 2 log3 and z = log a - log 5; find the value of a if x + y = 2z.

 

Solution 2

 

 

 

 

 

 

 

 

Question 3

If x = log 0.6; y = log 1.25 and z = log 3 - 2 log 2, find the values of:

(i) x+y- z        (ii) 5x + y - z

Solution 3

 

(i)

 

(ii)

 

Question 4

If a2 = log x, b3 = log y and 3a2 - 2b3 = 6 log z, express y in terms of x and z.

Solution 4

 

Question 5

If log (log a + log b), show that: a2 + b2 = 6ab.

Solution 5

 

 

Question 6

If a2 + b2 = 23ab, show that:

log (log a + log b).

 

Solution 6

 

Question 7

If m = log 20 and n = log 25, find the value of x, so that: 2 log (x - 4) = 2 m - n.

Solution 7

 

 

Question 8

Solve for x and y ; if x > 0 and y > 0;log xy = log + 2 log 2 = 2.

Solution 8

Question 9

Find x, if:

(i) logx 625 = -4 

(ii) logx (5x - 6) = 2

 

Solution 9

(i)

 

 (ii)

Question 10

  

Solution 10

 

   

 

Question 11

Solution 11

 

 

 

Question 12

  

 

Solution 12

 

 

 

Question 13

Given log10x = 2a and log10y =  .

 

 

(i) Write 10a in terms of x.

 

 

(ii) Write 102b + 1 in terms of y.

 

 

(iii) If  , express P in terms of x and y.

 

Solution 13


 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Question 14

Solve:

log5(x + 1) - 1 = 1 + log5(x - 1).

Solution 14

 

 

 

 

Question 15

Solve for x, if:

  

 

Solution 15

 

 

 

Question 16

  

Solution 16

  

Question 17

  

Solution 17

  

Question 18

  

Solution 18

  

Question 19

  

Solution 19

  

Question 20

  

Solution 20

  

Question 21

Solution 21

Question 22

Solution 22

Chapter 8 - Logarithms Excercise Ex. 8(A)

Question 1

Express each of the following in logarithmic form:

(i) 53 = 125

(ii) 3-2 =

 

(iii) 10-3 = 0.001

 

(iv)

Solution 1

(i)

 

 

(ii)

 

 

(iii)

 

 

(iv)

open parentheses 81 close parentheses to the power of 3 over 4 end exponent equals 27
rightwards double arrow log subscript 81 27 equals 3 over 4 space left square bracket By space definition space of space logarithm comma space straight a to the power of straight b equals straight c rightwards double arrow log subscript straight a straight c equals straight b right square bracket 

Question 2

Express each of the following in exponential form:

(i) logg 0.125 = -1

(ii) log100.01 = -2

(iii) logaA = x

(iv) log101 = 0

Solution 2

(i)

(ii)

(iii)

(iv)

Question 3

Solve for x: log10 x = -2.

Solution 3

Question 4

Find the logarithm of:

(i) 100 to the base 10

(ii) 0.1 to the base 10

(iii) 0.001 to the base 10

(iv) 32 to the base 4

(v) 0.125 to the base 2

(vi) to the base 4

(vii) 27 to the base 9

(viii) to the base 27

Solution 4

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Question 5

State, true or false:

(i) If log10 x = a, then 10x = a.

(ii) If xy = z, then y = logzx.

(iii) log2 8 = 3 and log8 = 2 = .

Solution 5

(i)

(ii)

(iii)

Question 6

Find x, if:

(i) log3 x = 0

(ii) logx 2 = -1

(iii) log9243 = x

(iv) log5 (x - 7) = 1

(v) log432 = x - 4

(vi) log7 (2x2 - 1) = 2

Solution 6

(i)

 

(ii)

 

(iii)

 

 

(iv)

 

 

(v)

 

(vi)

 

 

Question 7

Evaluate:

(i) log10 0.01

(ii) log2 (1 ÷ 8)

(iii) log5 1

(iv) log5 125

(v) log16 8

(vi) log0.5 16

 

Solution 7

(i)

 

 

(ii)

 

 

(iii)

 

 

(iv)

 

 

(v)

 

 

(vi)

 

Question 8

If loga m = n, express an - 1 in terms in terms of a and m.

Solution 8

  

Question 9

  

Solution 9

  

Question 10

  

Solution 10

Question 11

  

Solution 11

  

Question 12

If log (x2 - 21) = 2, show that x = ± 11.

Solution 12

begin mathsize 14px style log space open parentheses straight x squared minus 21 close parentheses equals 2
rightwards double arrow straight x squared minus 21 equals 10 squared
rightwards double arrow straight x squared minus 21 equals 100
rightwards double arrow straight x squared equals 121
rightwards double arrow straight x equals plus-or-minus 11 end style

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