ICSE Class 9 Logarithms Solution New Pattern By Clarify Knowledge

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ICSE Class 9 Logarithms Solution Table

Chapter 8 - Logarithms Exercise 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)

Question 2

Express each of the following in exponential form:

(i) log_g 0.125 = -1

(ii) log₁₀0.01 = -2

(iii) log_aA = x

(iv) log₁₀1 = 0Solution 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 27Solution 4

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

Question 5

State, true or false:

(i) If log₁₀ x = a, then 10^x = a.

(ii) If x^y = z, then y = log_zx.

(iii) log₂ 8 = 3 and log₈ = 2 = .Solution 5

(i)

(ii)

(iii)

Question 6

Find x, if:

(i) log₃ x = 0

(ii) log_x 2 = -1

(iii) log₉243 = x

(iv) log₅ (x - 7) = 1

(v) log₄32 = x - 4

(vi) log₇ (2x² - 1) = 2Solution 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) log_0.5 16Solution 7

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 8

If log_a m = n, express a^{n - 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

Chapter 8 - Logarithms Exercise 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 3Solution 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 32Solution 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 3Solution 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 ySolution 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 8Solution 12

(i)

(ii)

(iii)

(iv)

(v)

(vi)

Question 13

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

(i) x(ii) log₁₀ 2xSolution 13

(i)

(ii)

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Chapter 8 - Logarithms Exercise Ex. 8(C)

Question 1

If log₁₀ 8 = 0.90; find the value of:

(i) log₁₀ 4(ii) log 

(iii) log 0.125Solution 1

(i)

(ii)

(iii)

Question 2

If log 27 = 1.431, find the value of :

(i) log 9(ii) log 300Solution 2

(i)

(ii)

Question 3

If log₁₀ a = b, find 10^{3b - 2} in terms of a.Solution 3

Question 4

If log₅ x = y, find 5^{2y+ 3} in terms of x.Solution 4

Question 5

Given: log₃ m = x and log₃ n = y.

(i) Express 3^{2x - 3} in terms of m.

(ii) Write down 3^{1 - 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)

Question 6

Simplify:

(i) log (a)³ - log a

(ii) log (a)³  log aSolution 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 b^a. a^b = 10Solution 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

Question 11

Solution 11

Chapter 8 - Logarithms Exercise 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) 5^{x + 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) log_x 625 = -4 

(ii) log_x (5x - 6) = 2Solution 9

(i)

 (ii)

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Given log₁₀x = 2a and log₁₀y = .

(i) Write 10^a in terms of x.

(ii) Write 10^{2b + 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