ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms

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) 7^x =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:

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) log_a ((frac { 1 }{ a })) =-1

Solution:

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:

Question 4.

Solve the following equations for x.



Solution:

Question 5.

Given log₁₀a = b, express 10^2b-3 in terms of a.

Solution:

Question 6.

Given log₁₀ x= a, log₁₀ y = b and log₁₀ z =c,

(i) write down 10^2a-3 in terms of x.

(ii) write down 10^3b-1 in terms of y.

(iii) if log₁₀ P = 2a + (frac { b }{ 2 })– 3c, express P in terms of x, y and z.

Solution:

Question 7.

If log₁₀x = a and log₁₀y = b, find the value of xy.

Solution:

Question 8.

Given log₁₀ a = m and log₁₀ b = n, express (frac { { a }^{ 3 } }{ { b }^{ 2 } }) in terms of m and n.

Solution:

Question 9.

Given log₁₀a= 2a and log₁₀y = –(frac { b }{ 2 })

(i) write 10^a in terms of x.

(ii) write 10^2b+1 in terms of y.

(iii) if log₁₀P= 3a -2b, express P in terms of x and y .

Solution:

Question 10.

If log₂ y = x and log₃ z = x, find 72^x in terms of y and z.

Solution:

Question 11.

If log₂ x = a and log₅y = a, write 100^2a-1 in terms of x and y.

Solution:

Exercise 9.2

Question 1.

Simplify the following :



Solution:

Question 2.

Evaluate the following:





Solution:

Question 3.

Express each of the following as a single logarithm:



Solution:

Question 4.

Prove the following :

(i) log₁₀ 4 ÷ log₁₀ 2 = l0g₃ 9

(ii) log₁₀ 25 + log₁₀ 4 = log₅ 25

Solution:

Question 5.

If x = 100)^a , y = (10000)^b and z = (10)^c, express



Solution:

Question 6.

If a = log₁₀x, find the following in terms of a :

(i) x

(ii) log₁₀(sqrt [ 5 ]{ { x }^{ 2 } })

(iii) log₁₀5x

Solution:

Question 7.



Solution:

Question 8.



Solution:

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) 7^x-y-z

Solution:

Question 10.

If log V + log3 = log π + log4 + 3 log r, find V in terns of other quantities.

Solution:

Question 11.

Given 3 (log 5 – log3) – (log 5-2 log 6) = 2 – log n , find n.

Solution:

Question 12.

Given that log₁₀y + 2 log₁₀x= 2, express y in terms of x.

Solution:

Question 13.

Express log₁₀2+1 in the from log₁₀x.

Solution:

Question 14.



Solution:

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:

Question 16.



Solution:

Question 17.



Solution:

Question 18.

Solve for x:



Solution:

Question 19.

Given 2 log₁₀x+1= log₁₀250, find

(i) x

(ii) log₁₀2x

Solution:

Question 20.



Solution:

Question 21.

Prove the following :

(i) 3^{_{log 4}} = 4^{_{log 3}}

(ii) 27^{_{log 2}} = 8^{_{log 3}}

Solution:

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:

Question 23.

Solve for x :

log₃ (x + 1) – 1 = 3 + log₃ (x – 1)

Solution:

Question 24.



Solution:

Question 25.



Solution:

Question 26.



Solution:

Question 27.

If p = log₁₀20 and q = log₁₀25, find the value of x if 2 log₁₀ (x +1) = 2p – q.

Solution:

Question 28.

Show that:



Solution:

Question 29.

Prove the following identities:



Solution:

Question 30.



Solution:

Question 31.

Solve for x :



Solution:

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:

Question 2.

If log₅ (0.04) = x, then the vlaue of x is

(a) 2

(b) 4

(c) -4

(d) -2

Solution:

Question 3.

If log_0.5 64 = x, then the value of x is

(a) -4

(b) -6

(c) 4

(d) 6

Solution:

Question 4.

If log_{10(sqrt [ 3 ]{ 5 })} x = -3, then the value of x is



Solution:

Question 5.

If log (3x + 1) = 2, then the value of x is



Solution:

Question 6.

The value of 2 + log₁₀ (0.01) is

(a)4

(b)3

(c)1

(d)0

Solution:

Question 7.





Solution:

Chapter Test

Question 1.



Solution:

Question 2.

Find the value of log√3 3√3 – log₅ (0.04)

Solution:

Question 3.

Prove the following:



Solution:

Question 4.

If log (m + n) = log m + log n, show that n = (frac { m }{ m-1 })

Solution:

Question 5.



Solution:

Question 6.



Solution:

Question 7.

Solve the following equations for x:



Solution:

Question 8.

Solve for x and y:



Solution:

Question 9.

If a = 1 + log_xyz, 6 = 1+ log_y zx and c=1 + log_zxy, then show that ab + bc + ca = abc.

Solution: