ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms

Exercise 9.1

Question 1.
Convert the following to logarithmic form:
(i) 52 = 25
(ii) a5 =64
(iii) 7x =100
(iv) 9° = 1
(v) 61 = 6
(vi) 3-2 = \(\frac { 1 }{ 9 }\)
(vii) 10-2 = 0.01
(viii) (81)\(\frac { 3 }{ 4 }\) = 27
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q1.1

Question 2.
Convert the following into exponential form:
(i) log2 32 = 5
(ii) log3 81=4
(iii) log3\(\frac { 1 }{ 3 }\)= -1
(iv) log3 4= \(\frac { 2 }{ 3 }\)
(v) log8 32= \(\frac { 5 }{ 3 }\)
(vi) log10 (0.001) = -3
(Vii) log2 0.25 = -2
(viii) loga (\(\frac { 1 }{ a }\)) =-1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q2.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.1 Q2

Question 3.
By converting to exponential form, find the values of:
(i) log2 16
(ii) log5 125
(iii) log4 8
(iv) log9 27
(v) log10(.01)
(vi) log7 \(\frac { 1 }{ 7 }\)
(vii) log5 256
(Viii) log2 0.25
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q3.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q3.2

Question 4.
Solve the following equations for x.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q4.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q4.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q4.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q4.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q4.5

Question 5.
Given log10a = b, express 102b-3 in terms of a.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q5.1

Question 6.
Given log10 x= a, log10 y = b and log10 z =c,
(i) write down 102a-3 in terms of x.
(ii) write down 103b-1 in terms of y.
(iii) if log10 P = 2a + \(\frac { b }{ 2 }\)– 3c, express P in terms of x, y and z.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q6.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q6.2

Question 7.
If log10x = a and log10y = b, find the value of xy.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q7.1

Question 8.
Given log10 a = m and log10 b = n, express \(\frac { { a }^{ 3 } }{ { b }^{ 2 } }\) in terms of m and n.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q8.1

Question 9.
Given log10a= 2a and log10y = –\(\frac { b }{ 2 }\)
(i) write 10a in terms of x.
(ii) write 102b+1 in terms of y.
(iii) if log10P= 3a -2b, express P in terms of x and y .
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q9.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q9.2

Question 10.
If log2 y = x and log3 z = x, find 72x in terms of y and z.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q10.1

Question 11.
If log2 x = a and log5y = a, write 1002a-1 in terms of x and y.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Q11.1

Exercise 9.2

Question 1.
Simplify the following :
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q1.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q1.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q1.3

Question 2.
Evaluate the following:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q2.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q2.2
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q2.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q2.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q2.5
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q2.6

Question 3.
Express each of the following as a single logarithm:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q3.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q3.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q3.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q3.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q3.5

Question 4.
Prove the following :
(i) log10 4 ÷ log10 2 = l0g3 9
(ii) log10 25 + log10 4 = log5 25
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q4.1

Question 5.
If x = 100)a , y = (10000)b and z = (10)c, express
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q5.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q5.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q5.3

Question 6.
If a = log10x, find the following in terms of a :
(i) x
(ii) log10\(\sqrt [ 5 ]{ { x }^{ 2 } }\)
(iii) log105x
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q6.1

Question 7.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q7.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q7.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q7.3

Question 8.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q8.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q8.2

Question 9.
If x = log10 12, y = log4 2 x log10 9 and z = log10 0.4, find the values of
(i)x-y-z
(ii) 7x-y-z
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q9.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q9.2

Question 10.
If log V + log3 = log π + log4 + 3 log r, find V in terns of other quantities.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q10.1

Question 11.
Given 3 (log 5 – log3) – (log 5-2 log 6) = 2 – log n , find n.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q11.1

Question 12.
Given that log10y + 2 log10x= 2, express y in terms of x.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q12.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q12.2

Question 13.
Express log102+1 in the from log10x.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q13.1

Question 14.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q14.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q14.2

Question 15.
Given that log m = x + y and log n = x-y, express the value of log m²n in terms of x and y.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q15.1

Question 16.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q16.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q16.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q16.3

Question 17.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q17.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q17.2

Question 18.
Solve for x:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q18.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q18.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q18.3

Question 19.
Given 2 log10x+1= log10250, find
(i) x
(ii) log102x
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q19.1

Question 20.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q20.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q20.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q20.3

Question 21.
Prove the following :
(i) 3log 4 = 4log 3
(ii) 27log 2 = 8log 3

Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q21.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q21.2

Question 22.
Solve the following equations :
(i) log (2x + 3) = log 7
(ii) log (x +1) + log (x – 1) = log 24
(iii) log (10x + 5) – log (x – 4) = 2
(iv) log105 + log10(5x+1) = log10(x + 5) + 1
(v) log (4y – 3) = log (2y + 1) – log3
(vi) log10(x + 2) + log10(x – 2) = log103 + 31og104.
(vii) log(3x + 2) + log(3x – 2) = 5 log 2.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q22.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q22.2

Question 23.
Solve for x :
log3 (x + 1) – 1 = 3 + log3 (x – 1)
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q23.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q23.2

Question 24.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms ex Q24
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q24.1

Question 25.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q25.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q25.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q25.3

Question 26.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q26.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q26.2

Question 27.
If p = log1020 and q = log1025, find the value of x if 2 log10 (x +1) = 2p – q.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q27.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q27.2

Question 28.
Show that:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q28.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q28.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q28.3

Question 29.
Prove the following identities:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q29.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q29.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q29.3

Question 30.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q30.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q30.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q30.3

Question 31.
Solve for x :
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q31.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q31.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 Q31.3

Multiple Choice Questions

correct Solution from the given four options (1 to 7):
Question 1.
If log√3 27 = x, then the value of x is
(a) 3
(b) 4
(c) 6
(d) 9
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q1.1

Question 2.
If log5 (0.04) = x, then the vlaue of x is
(a) 2
(b) 4
(c) -4
(d) -2
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q2.1

Question 3.
If log0.5 64 = x, then the value of x is
(a) -4
(b) -6
(c) 4
(d) 6
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q3.1

Question 4.
If log10\(\sqrt [ 3 ]{ 5 }\) x = -3, then the value of x is
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms Mcq 4
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q4.1

Question 5.
If log (3x + 1) = 2, then the value of x is
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q5.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q5.2

Question 6.
The value of 2 + log10 (0.01) is
(a)4
(b)3
(c)1
(d)0
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q6.1

Question 7.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q7.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q7.2
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 mul Q7.3

Chapter Test

Question 1.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q1.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q1.2

Question 2.
Find the value of log√3 3√3 – log5 (0.04)
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q2.1
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q2.2

Question 3.
Prove the following:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q3.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q3.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q3.3

Question 4.
If log (m + n) = log m + log n, show that n = \(\frac { m }{ m-1 }\)
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q4.1

Question 5.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q5.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q5.2

Question 6.
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q6.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q6.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q6.3

Question 7.
Solve the following equations for x:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q7.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q7.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q7.3
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q7.4
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q7.5

Question 8.
Solve for x and y:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q8.1
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q8.2
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q8.3

Question 9.
If a = 1 + logxyz, 6 = 1+ logy zx and c=1 + logzxy, then show that ab + bc + ca = abc.
Solution:
ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms 9.2 ch Q9.1

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