LOGARITHMS - Questions & Answers

EXERCISE 8(A)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) The value of log_√2 8 is :
(i) 6 (ii) 4 (iii) 3 (iv) 8
Step 1: Let log_√2 8 = x
Step 2: (√2)^x = 8
Step 3: (2^(1/2))^x = 2^3
Step 4: 2^(x/2) = 2^3
Step 5: x/2 = 3 => x = 6
Answer: (i) 6

(b) If log_4 x = 2.5, the value of x is :
(i) 12.5 (ii) 32 (iii) 10 (iv) 20
Step 1: log_4 x = 2.5
Step 2: 4^2.5 = x
Step 3: (2^2)^2.5 = x
Step 4: 2^5 = x => x = 32
Answer: (ii) 32

(c) If log_√3 x = 4, the value of x is :
(i) 12 (ii) 6 (iii) 9 (iv) 24
Step 1: log_√3 x = 4
Step 2: (√3)^4 = x
Step 3: (3^(1/2))^4 = x
Step 4: 3^2 = x => x = 9
Answer: (iii) 9

(d) If log_x 64 = 1.5, the value of x is :
(i) 48 (ii) 32 (iii) 64 (iv) 16
Step 1: log_x 64 = 1.5 = 3/2
Step 2: x^(3/2) = 64
Step 3: (x^(1/2))^3 = 4^3
Step 4: x^(1/2) = 4 => x = 16
Answer: (iv) 16

(e) If log_2 (x^2 - 4) = 5, the value of x is :
(i) ± 6 (ii) 6 (iii) - 6 (iv) ± 12
Step 1: log_2 (x^2 - 4) = 5
Step 2: 2^5 = x^2 - 4
Step 3: 32 = x^2 - 4
Step 4: 36 = x^2 => x = ± 6
Answer: (i) ± 6

(f) If log_10 x = a, the value of 10^(a-1) in terms of x is :
(i) 10x (ii) x/10 (iii) 10/x (iv) 1/10x
Step 1: log_10 x = a => 10^a = x
Step 2: 10^(a-1) = 10^a / 10^1
Step 3: Substitute 10^a = x
Step 4: 10^(a-1) = x / 10
Answer: (ii) x/10

2. Express each of the following in logarithmic form :
(i) 5^3 = 125
Answer: log_5 125 = 3
(ii) 3^-2 = 1/9
Answer: log_3 (1/9) = -2
(iii) 10^-3 = 0.001
Answer: log_10 0.001 = -3
(iv) (81)^(3/4) = 27
Answer: log_81 27 = 3/4

3. Express each of the following in exponential form :
(i) log_8 0.125 = -1
Answer: 8^-1 = 0.125
(ii) log_10 0.01 = -2
Answer: 10^-2 = 0.01
(iii) log_a A = x
Answer: a^x = A
(iv) log_10 1 = 0
Answer: 10^0 = 1

4. Solve for x : log_10 x = -2.
Step 1: Convert to exponential form.
Step 2: x = 10^-2
Step 3: x = 1/100 or 0.01
Answer: x = 0.01

5. Find the logarithm of :
(i) 100 to the base 10
Step 1: Let log_10 100 = x
Step 2: 10^x = 100 = 10^2
Step 3: x = 2
Answer: 2
(ii) 0.1 to the base 10
Step 1: Let log_10 0.1 = x
Step 2: 10^x = 0.1 = 10^-1
Step 3: x = -1
Answer: -1
(iii) 0.001 to the base 10
Step 1: Let log_10 0.001 = x
Step 2: 10^x = 10^-3
Step 3: x = -3
Answer: -3
(iv) 32 to the base 4
Step 1: Let log_4 32 = x
Step 2: 4^x = 32 => (2^2)^x = 2^5
Step 3: 2^(2x) = 2^5 => 2x = 5 => x = 5/2
Answer: 5/2
(v) 0.125 to the base 2
Step 1: Let log_2 0.125 = x
Step 2: 2^x = 1/8 = 2^-3
Step 3: x = -3
Answer: -3
(vi) 1/16 to the base 4
Step 1: Let log_4 (1/16) = x
Step 2: 4^x = 1/16 = 4^-2
Step 3: x = -2
Answer: -2
(vii) 27 to the base 9
Step 1: Let log_9 27 = x
Step 2: 9^x = 27 => (3^2)^x = 3^3
Step 3: 2x = 3 => x = 3/2
Answer: 3/2
(viii) 1/81 to the base 27
Step 1: Let log_27 (1/81) = x
Step 2: 27^x = 1/81 => (3^3)^x = 3^-4
Step 3: 3x = -4 => x = -4/3
Answer: -4/3

6. State, true or false :
(i) If log_10 x = a, then 10^x = a.
Step 1: By definition, log_10 x = a means 10^a = x.
Step 2: The given statement is 10^x = a, which is incorrect.
Answer: False
(ii) If x^y = z, then y = log_z x.
Step 1: By definition, x^y = z means log_x z = y.
Step 2: The statement says log_z x = y, which is incorrect.
Answer: False
(iii) log_2 8 = 3 and log_8 2 = 1/3.
Step 1: log_2 8 = x => 2^x = 2^3 => x=3 (True).
Step 2: log_8 2 = y => 8^y = 2 => 2^(3y) = 2^1 => 3y=1 => y=1/3 (True).
Answer: True

7. Find x, if :
(i) log_3 x = 0
Step 1: 3^0 = x
Step 2: x = 1
Answer: x = 1
(ii) log_x 2 = -1
Step 1: x^-1 = 2
Step 2: 1/x = 2 => x = 1/2
Answer: x = 1/2
(iii) log_9 243 = x
Step 1: 9^x = 243
Step 2: (3^2)^x = 3^5
Step 3: 2x = 5 => x = 5/2
Answer: x = 5/2
(iv) log_5 (x - 7) = 1
Step 1: 5^1 = x - 7
Step 2: 5 + 7 = x => x = 12
Answer: x = 12
(v) log_4 32 = x - 4
Step 1: 4^(x - 4) = 32
Step 2: (2^2)^(x-4) = 2^5
Step 3: 2(x - 4) = 5 => 2x - 8 = 5 => 2x = 13 => x = 13/2
Answer: x = 13/2
(vi) log_7 (2x^2 - 1) = 2
Step 1: 7^2 = 2x^2 - 1
Step 2: 49 = 2x^2 - 1
Step 3: 50 = 2x^2 => x^2 = 25 => x = ±5
Answer: x = 5 or x = -5

8. Evaluate :
(i) log_10 0.01
Step 1: log_10 (10^-2)
Step 2: -2 log_10 10 = -2
Answer: -2
(ii) log_2 (1 ÷ 8)
Step 1: log_2 (1/8) = log_2 (2^-3)
Step 2: -3
Answer: -3
(iii) log_5 1
Step 1: log of 1 to any base is 0.
Answer: 0
(iv) log_5 125
Step 1: log_5 (5^3)
Step 2: 3
Answer: 3
(v) log_16 8
Step 1: Let log_16 8 = x
Step 2: 16^x = 8 => 2^(4x) = 2^3
Step 3: 4x = 3 => x = 3/4
Answer: 3/4
(vi) log_0.5 16
Step 1: Let log_0.5 16 = x
Step 2: (1/2)^x = 16 => 2^-x = 2^4
Step 3: -x = 4 => x = -4
Answer: -4

9. If log_a m = n, express a^(n - 1) in terms of a and m.
Step 1: log_a m = n implies a^n = m.
Step 2: We need a^(n - 1) = a^n / a^1.
Step 3: Substitute a^n = m to get m / a.
Answer: m / a

10. Given log_2 x = m and log_5 y = n.
(i) Express 2^(m - 3) in terms of x.
Step 1: log_2 x = m implies 2^m = x.
Step 2: 2^(m - 3) = 2^m / 2^3.
Step 3: Substitute 2^m = x to get x / 8.
Answer: x / 8
(ii) Express 5^(3n + 2) in terms of y.
Step 1: log_5 y = n implies 5^n = y.
Step 2: 5^(3n + 2) = (5^n)^3 * 5^2.
Step 3: Substitute 5^n = y to get y^3 * 25.
Answer: 25y^3

EXERCISE 8(B)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) The value of 3 + log_5 5^-2 is :
(i) 1 (ii) 5 (iii) 3 - 1/5 (iv) 3 + 1/5
Step 1: 3 + log_5 5^-2
Step 2: 3 + (-2)log_5 5
Step 3: 3 - 2(1) = 1
Answer: (i) 1

(b) The value of log_5 75 - log_5 3 is :
(i) 72 (ii) 2 (iii) 25 (iv) 5
Step 1: log_5 (75 / 3) = log_5 25
Step 2: log_5 (5^2) = 2 log_5 5 = 2
Answer: (ii) 2

(c) The value of log_5 125 ÷ log_5 √5 is :
(i) 5 (ii) 120 (iii) 6 (iv) 60
Step 1: log_5 (5^3) ÷ log_5 (5^(1/2))
Step 2: 3 ÷ (1/2)
Step 3: 3 * 2 = 6
Answer: (iii) 6

(d) The value of (√x)^(4 log_x a) is :
(i) a (ii) ax (iii) 1/2ax (iv) a^2
Step 1: (x^(1/2))^(4 log_x a)
Step 2: x^( (1/2) * 4 log_x a ) = x^(2 log_x a)
Step 3: x^(log_x a^2)
Step 4: Using property a^(log_a m) = m, this evaluates to a^2.
Answer: (iv) a^2

(e) If log 125 / log (1/5) = log x, the value of x is :
(i) 0.001 (ii) 0.01 (iii) 25 (iv) 5
Step 1: log (5^3) / log (5^-1) = log x
Step 2: 3 log 5 / (-1 log 5) = log x
Step 3: -3 = log_10 x
Step 4: x = 10^-3 = 0.001
Answer: (i) 0.001

(f) If log (x - 5) + log (x + 5) = 2 log 12, the positive value of x is :
(i) 5 (ii) 13 (iii) 4.8 (iv) 12
Step 1: log ((x - 5)(x + 5)) = log (12^2)
Step 2: x^2 - 25 = 144
Step 3: x^2 = 169 => x = 13 (since x > 0)
Answer: (ii) 13

2. Express in terms of log 2 and log 3 :
(i) log 36
Step 1: log (2^2 * 3^2)
Step 2: 2 log 2 + 2 log 3
Answer: 2 log 2 + 2 log 3
(ii) log 144
Step 1: log (2^4 * 3^2)
Step 2: 4 log 2 + 2 log 3
Answer: 4 log 2 + 2 log 3
(iii) log 4.5
Step 1: log (9/2) = log 3^2 - log 2
Step 2: 2 log 3 - log 2
Answer: 2 log 3 - log 2
(iv) log 26/51 - log 91/119
Step 1: log ( (26/51) * (119/91) )
Step 2: (2 * 13 * 7 * 17) / (3 * 17 * 7 * 13) = 2/3.
Step 3: log (2/3) = log 2 - log 3
Answer: log 2 - log 3
(v) log 75/16 - 2 log 5/9 + log 32/243
Step 1: log (75/16) - log (25/81) + log (32/243)
Step 2: log [ (75/16) * (81/25) * (32/243) ]
Step 3: (3*25 * 81 * 32) / (16 * 25 * 3*81) = 32 / 16 = 2
Step 4: log 2
Answer: log 2

3. Express each of the following in a form free from logarithm :
(i) 2 log x - log y = 1
Step 1: log x^2 - log y = log 10
Step 2: log (x^2 / y) = log 10
Step 3: x^2 / y = 10 => x^2 = 10y
Answer: x^2 = 10y
(ii) 2 log x + 3 log y = log a
Step 1: log x^2 + log y^3 = log a
Step 2: log (x^2 * y^3) = log a
Step 3: x^2 y^3 = a
Answer: x^2 y^3 = a
(iii) a log x - b log y = 2 log 3
Step 1: log x^a - log y^b = log 3^2
Step 2: log (x^a / y^b) = log 9
Step 3: x^a / y^b = 9
Answer: x^a / y^b = 9

4. Evaluate each of the following :
(i) log 5 + log 8 - 2 log 2
Step 1: log 5 + log 8 - log 4
Step 2: log ( (5 * 8) / 4 ) = log 10
Step 3: 1
Answer: 1
(ii) log_10 8 + log_10 25 + 2 log_10 3 - log_10 18
Step 1: log_10 8 + log_10 25 + log_10 9 - log_10 18
Step 2: log_10 [ (8 * 25 * 9) / 18 ]
Step 3: log_10 ( 1800 / 18 ) = log_10 100
Step 4: 2
Answer: 2
(iii) log 4 + 1/3 log 125 - 1/5 log 32
Step 1: log 4 + log (125^(1/3)) - log (32^(1/5))
Step 2: log 4 + log 5 - log 2
Step 3: log ( (4 * 5) / 2 ) = log 10
Step 4: 1
Answer: 1

5. Prove that :
2 log 15/18 - log 25/162 + log 4/9 = log 2.
Step 1: LHS = log (15/18)^2 - log (25/162) + log (4/9)
Step 2: (15/18)^2 = (5/6)^2 = 25/36.
Step 3: log [ (25/36) * (162/25) * (4/9) ]
Step 4: [ 162 * 4 ] / [ 36 * 9 ] = 648 / 324 = 2.
Step 5: log 2 = RHS.
Answer: Hence Proved.

6. Find x, if :
x - log 48 + 3 log 2 = 1/3 log 125 - log 3.
Step 1: x - log 48 + log 8 = log (125^(1/3)) - log 3
Step 2: x - (log 48 - log 8) = log 5 - log 3
Step 3: x - log (48/8) = log (5/3)
Step 4: x - log 6 = log (5/3)
Step 5: x = log (5/3) + log 6 = log ( (5/3) * 6 ) = log 10
Step 6: x = 1
Answer: x = 1

7. Express log_10 2 + 1 in the form of log_10 x.
Step 1: log_10 2 + log_10 10
Step 2: log_10 (2 * 10)
Step 3: log_10 20
Answer: log_10 20

8. Solve for x :
(i) log_10 (x - 10) = 1
Step 1: x - 10 = 10^1
Step 2: x = 20
Answer: x = 20
(ii) log (x^2 - 21) = 2
Step 1: x^2 - 21 = 10^2 = 100
Step 2: x^2 = 121 => x = 11, -11
Answer: x = 11, x = -11
(iii) log (x - 2) + log (x + 2) = log 5
Step 1: log (x^2 - 4) = log 5
Step 2: x^2 - 4 = 5 => x^2 = 9
Step 3: x = 3 (since x > 2 for real logs)
Answer: x = 3
(iv) log (x + 5) + log (x - 5) = 4 log 2 + 2 log 3
Step 1: log (x^2 - 25) = log (2^4) + log (3^2)
Step 2: log (x^2 - 25) = log 16 + log 9 = log 144
Step 3: x^2 - 25 = 144 => x^2 = 169
Step 4: x = 13 (since x > 5)
Answer: x = 13

9. Solve for x :
(i) log 81 / log 27 = x
Step 1: x = log (3^4) / log (3^3)
Step 2: x = (4 log 3) / (3 log 3) = 4/3
Answer: x = 4/3
(ii) log 128 / log 32 = x
Step 1: x = log (2^7) / log (2^5)
Step 2: x = (7 log 2) / (5 log 2) = 7/5
Answer: x = 7/5
(iii) log 64 / log 8 = log x
Step 1: (6 log 2) / (3 log 2) = log x
Step 2: 2 = log x => x = 10^2 = 100
Answer: x = 100
(iv) log 225 / log 15 = log x
Step 1: log (15^2) / log 15 = log x
Step 2: 2 log 15 / log 15 = log x
Step 3: 2 = log x => x = 10^2 = 100
Answer: x = 100

10. Given log x = m + n and log y = m - n, express the value of log (10x / y^2) in terms of m and n.
Step 1: log (10x / y^2) = log 10 + log x - 2 log y
Step 2: 1 + (m + n) - 2(m - n)
Step 3: 1 + m + n - 2m + 2n = 1 - m + 3n
Answer: 1 - m + 3n

11. State, true or false :
(i) log 1 x log 1000 = 0
Step 1: log 1 = 0. 0 x log 1000 = 0.
Answer: True
(ii) log x / log y = log x - log y
Step 1: quotient of logs is not log of quotient. The statement is incorrect.
Answer: False
(iii) If log 25 / log 5 = log x, then x = 2
Step 1: (2 log 5) / (log 5) = 2 = log x.
Step 2: log x = 2 => x = 100, not 2.
Answer: False
(iv) log x × log y = log x + log y
Step 1: log (x * y) = log x + log y. The given is product of logs, which is false.
Answer: False

12. If log_10 2 = a and log_10 3 = b; express each of the following in terms of 'a' and 'b' :
(i) log 12
Step 1: log (4 * 3) = log 2^2 + log 3 = 2 log 2 + log 3
Answer: 2a + b
(ii) log 2.25
Step 1: log (225 / 100) = log (9 / 4) = log 3^2 - log 2^2
Step 2: 2 log 3 - 2 log 2
Answer: 2b - 2a
(iii) log 2 1/4
Step 1: log (9/4)
Step 2: log 9 - log 4 = 2 log 3 - 2 log 2
Answer: 2b - 2a
(iv) log 5.4
Step 1: log (54 / 10) = log 54 - 1
Step 2: log (2 * 27) - 1 = log 2 + 3 log 3 - 1
Answer: a + 3b - 1
(v) log 60
Step 1: log (6 * 10) = log 2 + log 3 + 1
Answer: a + b + 1
(vi) log 3 1/8
Step 1: log (25 / 8) = log (100 / 32) = log 10^2 - log 2^5
Step 2: 2 - 5 log 2
Answer: 2 - 5a

13. If log 2 = 0.3010 and log 3 = 0.4771; find the value of :
(i) log 12
Step 1: log (2^2 * 3) = 2 log 2 + log 3
Step 2: 2(0.3010) + 0.4771 = 0.6020 + 0.4771
Answer: 1.0791
(ii) log 1.2
Step 1: log (12 / 10) = log 12 - 1
Step 2: 1.0791 - 1
Answer: 0.0791
(iii) log 3.6
Step 1: log (36 / 10) = log (2^2 * 3^2) - 1 = 2 log 2 + 2 log 3 - 1
Step 2: 0.6020 + 0.9542 - 1 = 1.5562 - 1
Answer: 0.5562
(iv) log 15
Step 1: log (30 / 2) = log 3 + log 10 - log 2
Step 2: 0.4771 + 1 - 0.3010
Answer: 1.1761
(v) log 25
Step 1: log (100 / 4) = 2 - 2 log 2
Step 2: 2 - 0.6020
Answer: 1.3980
(vi) 2/3 log 8
Step 1: (2/3) * 3 log 2 = 2 log 2
Step 2: 2(0.3010)
Answer: 0.6020

14. Given 2 log_10 x + 1 = log_10 250, find :
(i) x
Step 1: log x^2 + log 10 = log 250
Step 2: log (10x^2) = log 250
Step 3: 10x^2 = 250 => x^2 = 25 => x = 5 (since x > 0)
Answer: x = 5
(ii) log_10 2x
Step 1: log_10 (2 * 5) = log_10 10
Step 2: 1
Answer: 1

15. Given 3 log x + 1/2 log y = 2, express y in terms of x.
Step 1: log x^3 + log y^(1/2) = 2
Step 2: log (x^3 * y^(1/2)) = log 100
Step 3: x^3 * y^(1/2) = 100
Step 4: y^(1/2) = 100 / x^3
Step 5: y = 10000 / x^6
Answer: y = 10000 / x^6

EXERCISE 8(C)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) If log_2 (log_3 x) = 4, the value of x is :
(i) 3^16 (ii) 16^3 (iii) 3 x 16 (iv) 16 ÷ 3
Step 1: log_3 x = 2^4 = 16
Step 2: x = 3^16
Answer: (i) 3^16

(b) If log (5x - 4) - log (x + 1) = log 4, the value of x is :
(i) 6 (ii) 8 (iii) 4 (iv) 12
Step 1: log [ (5x - 4) / (x + 1) ] = log 4
Step 2: (5x - 4) / (x + 1) = 4
Step 3: 5x - 4 = 4x + 4 => x = 8
Answer: (ii) 8

(c) If log x - log (2x - 1) = 1, the value of x is :
(i) 19/10 (ii) 19 x 10 (iii) 10/19 (iv) 1 / (19 x 10)
Step 1: log [ x / (2x - 1) ] = 1 = log 10
Step 2: x / (2x - 1) = 10
Step 3: x = 20x - 10 => 19x = 10 => x = 10/19
Answer: (iii) 10/19

(d) If x = log 3/5, y = log 5/4 and z = 2 log √3/2, the value of x + y - z is :
(i) 1 (ii) -1 (iii) 2 (iv) 0
Step 1: x + y = log (3/5) + log (5/4) = log (3/4)
Step 2: z = log ((√3/2)^2) = log (3/4)
Step 3: x + y - z = log (3/4) - log (3/4) = 0
Answer: (iv) 0

(e) If log v + log 3 = log π + log 4 + 3 log r, the value of v in terms of r and other constants is :
(i) 4/3 πr^3 (ii) 4πr^2 (iii) πr^3 (iv) 4/3 πr^2
Step 1: log (3v) = log (π * 4 * r^3)
Step 2: 3v = 4πr^3
Step 3: v = (4/3)πr^3
Answer: (i) 4/3 πr^3

(f) If log y + 2 log x = 2, the value of y in terms of x is :
(i) x^2 ÷ 100 (ii) 100 ÷ x^2 (iii) x ÷ 10 (iv) 10 ÷ x
Step 1: log y + log x^2 = 2
Step 2: log (yx^2) = log 100
Step 3: yx^2 = 100 => y = 100 / x^2
Answer: (ii) 100 ÷ x^2

2. If log_10 8 = 0.90; find the value of : (HOTS)
(i) log_10 4
Step 1: log_10 (2^3) = 0.90 => 3 log_10 2 = 0.90 => log_10 2 = 0.30
Step 2: log_10 4 = log_10 (2^2) = 2 log_10 2 = 2(0.30)
Answer: 0.60
(ii) log √32
Step 1: log (32^(1/2)) = (1/2) log (2^5) = (5/2) log 2
Step 2: (5/2)(0.30) = 5 * 0.15 = 0.75
Answer: 0.75
(iii) log 0.125
Step 1: log (1/8) = log (2^-3) = -3 log 2
Step 2: -3(0.30) = -0.90
Answer: -0.90

3. If log 27 = 1.431, find the value of :
(i) log 9
Step 1: log 27 = log 3^3 = 3 log 3 = 1.431 => log 3 = 0.477
Step 2: log 9 = log 3^2 = 2 log 3 = 2(0.477)
Answer: 0.954
(ii) log 300
Step 1: log (3 * 100) = log 3 + log 100 = log 3 + 2
Step 2: 0.477 + 2 = 2.477
Answer: 2.477

4. If log_10 a = b, find 10^(3b - 2) in terms of a.
Step 1: b = log_10 a => 10^b = a
Step 2: 10^(3b - 2) = (10^b)^3 / 10^2
Step 3: a^3 / 100
Answer: a^3 / 100

5. If log_5 x = y, find 5^(2y + 3) in terms of x.
Step 1: y = log_5 x => 5^y = x
Step 2: 5^(2y + 3) = (5^y)^2 * 5^3
Step 3: x^2 * 125
Answer: 125x^2

6. Given: log_3 m = x and log_3 n = y.
(i) Express 3^(2x - 3) in terms of m.
Step 1: 3^x = m
Step 2: 3^(2x - 3) = (3^x)^2 / 3^3 = m^2 / 27
Answer: m^2 / 27
(ii) Write down 3^(1 - 2y + 3x) in terms of m and n.
Step 1: 3^x = m, 3^y = n
Step 2: 3^1 * (3^y)^-2 * (3^x)^3
Step 3: 3 * n^-2 * m^3 = 3m^3 / n^2
Answer: 3m^3 / n^2
(iii) If 2 log_3 A = 5x - 3y; find A in terms of m and n.
Step 1: 2 log_3 A = 5(log_3 m) - 3(log_3 n)
Step 2: log_3 A^2 = log_3 (m^5 / n^3)
Step 3: A^2 = m^5 / n^3 => A = √(m^5 / n^3)
Answer: √(m^5 / n^3)

7. Simplify :
(i) log (a)^3 - log a
Step 1: 3 log a - log a = 2 log a
Answer: 2 log a
(ii) log (a)^3 ÷ log a
Step 1: (3 log a) / (log a) = 3
Answer: 3

8. If log (a + b) = log a + log b, find a in terms of b.
Step 1: log (a + b) = log (ab)
Step 2: a + b = ab
Step 3: b = ab - a = a(b - 1)
Step 4: a = b / (b - 1)
Answer: a = b / (b - 1)

9. Prove that :
(i) (log a)^2 - (log b)^2 = log (a/b) . log (ab)
Step 1: LHS = (log a - log b)(log a + log b)
Step 2: = log (a/b) * log (ab) = RHS
Answer: Hence Proved.
(ii) If a log b + b log a - 1 = 0, then b^a . a^b = 10
Step 1: log b^a + log a^b = 1
Step 2: log (b^a * a^b) = log 10
Step 3: b^a * a^b = 10
Answer: Hence Proved.

10. (i) If log (a + 1) = log (4a - 3) - log 3; find a.
Step 1: log (a + 1) = log ((4a - 3) / 3)
Step 2: a + 1 = (4a - 3) / 3
Step 3: 3a + 3 = 4a - 3 => a = 6
Answer: a = 6
(ii) If 2 log y - log x - 3 = 0, express x in terms of y.
Step 1: log y^2 - log x = 3
Step 2: log (y^2 / x) = log 1000
Step 3: y^2 / x = 1000 => x = y^2 / 1000
Answer: x = y^2 / 1000
(iii) Prove that : log_10 125 = 3(1 - log_10 2).
Step 1: RHS = 3(log_10 10 - log_10 2) = 3 log_10 5
Step 2: log_10 (5^3) = log_10 125 = LHS.
Answer: Hence Proved.

EXERCISE 8(D)

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) The value of (log 8 - log 2) ÷ log 32 is :
(i) 2/5 (ii) 5/2 (iii) 2 (iv) 4
Step 1: log (8/2) / log (2^5)
Step 2: log 4 / 5 log 2
Step 3: 2 log 2 / 5 log 2 = 2/5
Answer: (i) 2/5

(b) If log_3 (x + 1) = 2, the value of x is :
(i) 9 (ii) 8 (iii) 3 (iv) 1/3
Step 1: x + 1 = 3^2
Step 2: x + 1 = 9 => x = 8
Answer: (ii) 8

(c) If 2 log x = log 250 - 1, the value of x is:
(i) 17 (ii) -5 (iii) 5 (iv) 10
Step 1: log x^2 = log 250 - log 10
Step 2: log x^2 = log 25
Step 3: x^2 = 25 => x = 5 (x>0)
Answer: (iii) 5

(d) If log_3 x - log_3 2 - 1 = 0; the value of x is :
(i) 3 (ii) -3 (iii) -6 (iv) 6
Step 1: log_3 x - log_3 2 = 1
Step 2: log_3 (x/2) = 1
Step 3: x/2 = 3^1 => x = 6
Answer: (iv) 6

(e) The value of 2 log 3 - 1/3 log 64 + log 12 is :
(i) log 27 (ii) 27 (iii) -27 (iv) -log 27
Step 1: log 3^2 - log (64^(1/3)) + log 12
Step 2: log 9 - log 4 + log 12
Step 3: log [ (9 * 12) / 4 ] = log (108 / 4) = log 27
Answer: (i) log 27

2. If 3/2 log a + 2/3 log b - 1 = 0, find the value of a^9 . b^4.
Step 1: multiply by 6: 9 log a + 4 log b - 6 = 0
Step 2: log a^9 + log b^4 = 6
Step 3: log (a^9 b^4) = log 10^6
Step 4: a^9 b^4 = 10^6
Answer: 10^6

3. If x = 1 + log 2 - log 5, y = 2 log 3 and z = log a - log 5; find the value of a, if x + y = 2z.
Step 1: x = log 10 + log 2 - log 5 = log (20/5) = log 4
Step 2: y = log 9
Step 3: z = log (a/5)
Step 4: x + y = log 4 + log 9 = log 36
Step 5: 2z = 2 log (a/5) = log ((a/5)^2) = log (a^2/25)
Step 6: log 36 = log (a^2 / 25) => a^2 / 25 = 36 => a^2 = 900 => a = 30
Answer: a = 30

4. If x = log 0.6; y = log 1.25 and z = log 3 - 2 log 2, find the values of :
(i) x + y - z
Step 1: x = log (6/10) = log (3/5)
Step 2: y = log (5/4)
Step 3: z = log 3 - log 4 = log (3/4)
Step 4: x + y - z = log (3/5) + log (5/4) - log (3/4)
Step 5: log [ (3/5 * 5/4) / (3/4) ] = log ( (3/4) / (3/4) ) = log 1 = 0
Answer: 0
(ii) 5^(x + y - z)
Step 1: 5^0 = 1
Answer: 1

5. If a^2 = log x, b^3 = log y and 3a^2 - 2b^3 = 6 log z, express y in terms of x and z.
Step 1: 3(log x) - 2(log y) = log z^6
Step 2: log x^3 - log y^2 = log z^6
Step 3: log (x^3 / y^2) = log z^6
Step 4: x^3 / y^2 = z^6 => y^2 = x^3 / z^6
Step 5: y = √(x^3 / z^6) = x^(3/2) / z^3
Answer: y = x^(3/2) / z^3

6. If log (a - b) / 2 = 1/2 (log a + log b), show that: a^2 + b^2 = 6ab.
Step 1: log ((a - b)/2) = (1/2) log (ab) = log (√(ab))
Step 2: (a - b)/2 = √(ab)
Step 3: Square both sides: (a^2 - 2ab + b^2) / 4 = ab
Step 4: a^2 - 2ab + b^2 = 4ab
Step 5: a^2 + b^2 = 6ab
Answer: Hence shown.

7. If a^2 + b^2 = 23ab, show that : log (a + b) / 5 = 1/2 (log a + log b).
Step 1: a^2 + b^2 + 2ab = 25ab => (a + b)^2 = 25ab
Step 2: ((a + b)/5)^2 = ab
Step 3: Take log of both sides: log ((a + b)/5)^2 = log (ab)
Step 4: 2 log ((a + b)/5) = log a + log b
Step 5: log ((a + b)/5) = (1/2) (log a + log b)
Answer: Hence shown.

8. If m = log 20 and n = log 25, find the value of x, so that : 2 log (x - 4) = 2m - n.
Step 1: 2 log (x - 4) = 2 log 20 - log 25
Step 2: log (x - 4)^2 = log 400 - log 25 = log 16
Step 3: (x - 4)^2 = 16 => x - 4 = ±4
Step 4: x = 8 or x = 0. (For log(x-4), x > 4, so x = 8)
Answer: x = 8

9. Solve for x and y; if x > 0 and y > 0 : log xy = log x/y + 2 log 2 = 2.
Step 1: log xy = 2 => xy = 100
Step 2: log (x/y) + log 4 = 2 => log (4x/y) = 2 => 4x/y = 100 => x = 25y
Step 3: Substitute x in eq 1: (25y)y = 100 => 25y^2 = 100 => y^2 = 4 => y = 2
Step 4: x = 25(2) = 50
Answer: x = 50, y = 2

10. Find x, if :
(i) log_x 625 = -4
Step 1: x^-4 = 625 = 5^4
Step 2: x^-4 = (1/5)^-4 => x = 1/5
Answer: x = 1/5
(ii) log_x (5x - 6) = 2.
Step 1: x^2 = 5x - 6
Step 2: x^2 - 5x + 6 = 0
Step 3: (x - 2)(x - 3) = 0 => x = 2, 3
Answer: x = 2, 3

11. If p = log 20 and q = log 25, find the value of x, if 2 log (x + 1) = 2p - q.
Step 1: 2 log (x + 1) = 2 log 20 - log 25
Step 2: log (x + 1)^2 = log (400 / 25) = log 16
Step 3: (x + 1)^2 = 16 => x + 1 = 4 => x = 3
Answer: x = 3

12. If log_2 (x + y) = log_3 (x - y) = log 25 / log 0.2, find the values of x and y.
Step 1: log 25 / log 0.2 = log 5^2 / log 5^-1 = 2 / -1 = -2.
Step 2: log_2 (x + y) = -2 => x + y = 2^-2 = 1/4
Step 3: log_3 (x - y) = -2 => x - y = 3^-2 = 1/9
Step 4: Add equations: 2x = 1/4 + 1/9 = 13/36 => x = 13/72
Step 5: Subtract equations: 2y = 1/4 - 1/9 = 5/36 => y = 5/72
Answer: x = 13/72, y = 5/72

13. Given : log x / log y = 3/2 and log (xy) = 5; find the values of x and y.
Step 1: log x = (3/2) log y => log x = log y^(3/2) => x = y^(3/2)
Step 2: log (xy) = 5 => xy = 10^5
Step 3: y^(3/2) * y = 10^5 => y^(5/2) = 10^5 => y = (10^5)^(2/5) = 10^2 = 100
Step 4: x = 100^(3/2) = 10^3 = 1000
Answer: x = 1000, y = 100

14. Given log_10 x = 2a and log_10 y = b/2.
(i) Write 10^a in terms of x.
Step 1: log_10 x = 2a => 10^(2a) = x
Step 2: (10^a)^2 = x => 10^a = √x
Answer: √x
(ii) Write 10^(2b + 1) in terms of y.
Step 1: log_10 y = b/2 => 10^(b/2) = y => 10^b = y^2 => 10^(2b) = y^4
Step 2: 10^(2b + 1) = 10^(2b) * 10 = 10y^4
Answer: 10y^4
(iii) If log_10 P = 3a - 2b, express P in terms of x and y.
Step 1: log_10 P = 3(log_10 x / 2) - 2(2 log_10 y) = (3/2) log_10 x - 4 log_10 y
Step 2: log_10 P = log_10 (x^(3/2)) - log_10 (y^4)
Step 3: P = x^(3/2) / y^4
Answer: P = x^(3/2) / y^4

15. Solve : log_5 (x + 1) - 1 = 1 + log_5 (x - 1).
Step 1: log_5 (x + 1) - log_5 (x - 1) = 2
Step 2: log_5 ( (x + 1) / (x - 1) ) = 2
Step 3: (x + 1) / (x - 1) = 5^2 = 25
Step 4: x + 1 = 25x - 25
Step 5: 24x = 26 => x = 26/24 = 13/12
Answer: x = 13/12

TEST YOURSELF

1. Multiple Choice Type : Choose the correct answer from the options given below.
(a) The value of log_3 81 is :
(i) 4 (ii) -4 (iii) 1/4 (iv) -1/4
Step 1: log_3 (3^4) = 4
Answer: (i) 4

(b) The value of log_16 2 is :
(i) 4 (ii) -4 (iii) 1/4 (iv) -1/4
Step 1: log_16 (16^(1/4)) = 1/4
Answer: (iii) 1/4

(c) If log (3x - 2) = 2, then the value of x is:
(i) 34 (ii) 30 (iii) 17 (iv) none of these
Step 1: 3x - 2 = 10^2 = 100
Step 2: 3x = 102 => x = 34
Answer: (i) 34

(d) 2 + 1/2 log 9 - 2 log 5 is equal to :
(i) -log 12 (ii) log 24 (iii) log 12 (iv) log 18/25
Step 1: log 100 + log (9^(1/2)) - log (5^2)
Step 2: log 100 + log 3 - log 25
Step 3: log [ (100 * 3) / 25 ] = log 12
Answer: (iii) log 12

(e) 3 + log 10^-2 is equal to :
(i) 5 (ii) 1 (iii) -5 (iv) -1
Step 1: 3 - 2 log 10 = 3 - 2 = 1
Answer: (ii) 1

(f) Statement (1) : log_2 (x^2 - 4) = 5 => x = 6.
Statement (2) : x^2 - 4 = 2^5 => x^2 = 36 and x = ±6
(i) Both the statements are true. (ii) Both the statements are false. (iii) Statement 1 is true, and statement 2 is false. (iv) Statement 1 is false, and statement 2 is true.
Step 1: x can be 6 or -6. Statement 1 gives only 6, but technically x can be -6. Statement 2 gives both.
Answer: (iv) Statement 1 is false, and statement 2 is true.

(g) Statement (1) : log_3 x = a; then 9^a = 1/x^2.
Statement (2) : log_3 x = a => 3^a = x .:. 9^a = (3^2)^a = (3^a)^2 = x^2
(i) Both the statements are true. (ii) Both the statements are false. (iii) Statement 1 is true, and statement 2 is false. (iv) Statement 1 is false, and statement 2 is true.
Answer: (iv) Statement 1 is false, and statement 2 is true.

(h) Assertion (A) : log_√3 x = 4 => x = 9
Reason (R) : x = (√3)^4 = 3^2 = 9
(i) A is true, R is false. (ii) A is false, R is true. (iii) Both A and R are true and R is the correct reason for A. (iv) Both A and R are true and R is the incorrect reason for A.
Answer: (iii) Both A and R are true and R is the correct reason for A.

(i) Assertion (A) : log 2 = a and log 3 = b => 1 + log 12 = 2a + b
Reason (R) : 1 + log 12 = 1 + log (2 x 2 x 3) = 1 + 2 log 2 + log 3 = 1 + 2a + b
(i) A is true, R is false. (ii) A is false, R is true. (iii) Both A and R are true and R is the correct reason for A. (iv) Both A and R are true and R is the incorrect reason for A.
Answer: (iii) Both A and R are true and R is the correct reason for A.

2. If log_2 x = a and log_3 y = a, write 72^a in terms of x and y.
Step 1: 2^a = x, 3^a = y
Step 2: 72^a = (8 * 9)^a = (2^3 * 3^2)^a = (2^a)^3 * (3^a)^2
Step 3: x^3 y^2
Answer: x^3 y^2

3. Solve for x : log (x - 1) + log (x + 1) = log_2 1.
Step 1: log ((x - 1)(x + 1)) = 0 (since log_2 1 = 0)
Step 2: log (x^2 - 1) = 0
Step 3: x^2 - 1 = 10^0 = 1 => x^2 = 2 => x = √2 (since x > 1)
Answer: x = √2

4. If log (x^2 - 21) = 2, show that x = ± 11.
Step 1: x^2 - 21 = 10^2 = 100
Step 2: x^2 = 121
Step 3: x = ± 11
Answer: Hence shown.

5. If x = (100)^a, y = (10000)^b and z = (10)^c, find log (10√y / x^2 z^3) in terms of a, b and c.
Step 1: x = 10^(2a), y = 10^(4b), z = 10^c
Step 2: 10√y = 10 * y^(1/2) = 10^1 * (10^(4b))^(1/2) = 10 * 10^(2b) = 10^(1 + 2b)
Step 3: x^2 z^3 = (10^(2a))^2 * (10^c)^3 = 10^(4a) * 10^(3c) = 10^(4a + 3c)
Step 4: Expression = log ( 10^(1 + 2b) / 10^(4a + 3c) )
Step 5: log ( 10^(1 + 2b - 4a - 3c) ) = 1 + 2b - 4a - 3c
Answer: 1 + 2b - 4a - 3c

6. If 3(log 5 - log 3) - (log 5 - 2 log 6) = 2 - log x, find x.
Step 1: 3 log 5 - 3 log 3 - log 5 + 2 log 6 = 2 - log x
Step 2: 2 log 5 - 3 log 3 + 2 log (2*3) = 2 - log x
Step 3: 2 log 5 - 3 log 3 + 2 log 2 + 2 log 3 = 2 - log x
Step 4: 2 log 5 + 2 log 2 - log 3 = 2 - log x
Step 5: 2(log 5 + log 2) - log 3 = 2 - log x
Step 6: 2(log 10) - log 3 = 2 - log x
Step 7: 2(1) - log 3 = 2 - log x => 2 - log 3 = 2 - log x => log x = log 3
Step 8: x = 3
Answer: x = 3

7. Given log x = 2m - n, log y = n - 2m and log z = 3m - 2n, find in terms of m and n, the value of log (x^2 y^3 / z^4).
Step 1: log x^2 + log y^3 - log z^4 = 2 log x + 3 log y - 4 log z
Step 2: 2(2m - n) + 3(n - 2m) - 4(3m - 2n)
Step 3: 4m - 2n + 3n - 6m - 12m + 8n
Step 4: (4 - 6 - 12)m + (-2 + 3 + 8)n = -14m + 9n
Answer: 9n - 14m

8. Given log_x 25 - log_x 5 = 2 - log_x 1/125; find x.
Step 1: log_x (25/5) = 2 - log_x (5^-3)
Step 2: log_x 5 = 2 + 3 log_x 5
Step 3: -2 log_x 5 = 2 => log_x 5 = -1
Step 4: x^-1 = 5 => x = 1/5
Answer: x = 1/5

9. Solve for x, if : log_x 49 - log_x 7 + log_x 1/343 + 2 = 0.
Step 1: log_x 49 - log_x 7 + log_x (7^-3) = -2
Step 2: log_x (49 / 7 * 7^-3) = -2
Step 3: log_x (7 * 7^-3) = -2 => log_x (7^-2) = -2
Step 4: -2 log_x 7 = -2 => log_x 7 = 1 => x^1 = 7 => x = 7
Answer: x = 7

10. If a^2 = log x, b^3 = log y and a^2/2 - b^3/3 = log c, find c in terms of x and y.
Step 1: (log x)/2 - (log y)/3 = log c
Step 2: log x^(1/2) - log y^(1/3) = log c
Step 3: log ( x^(1/2) / y^(1/3) ) = log c
Step 4: c = x^(1/2) / y^(1/3) = √x / ∛y
Answer: c = √x / ∛y

11. Given x = log_10 12, y = log_4 2 x log_10 9 and z = log_10 0.4, find :
(i) x - y - z
Step 1: y = (1/2) * log_10 9 = log_10 3. z = log_10 0.4.
Step 2: log_10 12 - log_10 3 - log_10 0.4
Step 3: log_10 ( 12 / (3 * 0.4) ) = log_10 ( 12 / 1.2 ) = log_10 10 = 1
Answer: 1
(ii) 13^(x - y - z)
Step 1: 13^1 = 13
Answer: 13

12. Solve for x, log_x 15√5 = 2 - log_x 3√5.
Step 1: log_x 15√5 + log_x 3√5 = 2
Step 2: log_x (15√5 * 3√5) = 2
Step 3: log_x (45 * 5) = 2
Step 4: log_x 225 = 2 => x^2 = 225 => x = 15
Answer: x = 15

13. Evaluate :
(i) log_b a x log_c b x log_a c
Step 1: (log a / log b) * (log b / log c) * (log c / log a) = 1
Answer: 1
(ii) log_3 8 ÷ log_9 16
Step 1: (log 8 / log 3) / (log 16 / log 9)
Step 2: (3 log 2 / log 3) * (2 log 3 / 4 log 2)
Step 3: (3/1) * (2/4) = 6/4 = 3/2
Answer: 3/2
(iii) log_5 8 / (log_25 16 x log_100 10)
Step 1: log_5 8 = 3 log_5 2
Step 2: log_25 16 = log 16 / log 25 = 4 log 2 / 2 log 5 = 2 log_5 2
Step 3: log_100 10 = log_100 (100^(1/2)) = 1/2
Step 4: Denominator = 2 log_5 2 * (1/2) = log_5 2
Step 5: Numerator / Denominator = (3 log_5 2) / (log_5 2) = 3
Answer: 3

14. Show that : log_a m ÷ log_ab m = 1 + log_a b (HOTS)
Step 1: LHS = (log m / log a) ÷ (log m / log ab)
Step 2: (log m / log a) * (log ab / log m) = log ab / log a
Step 3: (log a + log b) / log a = 1 + (log b / log a) = 1 + log_a b
Answer: Hence shown.

15. If log_√27 x = 2 2/3, find x.
Step 1: 2 2/3 = 8/3.
Step 2: (√27)^(8/3) = x
Step 3: ((3^3)^(1/2))^(8/3) = 3^(3/2 * 8/3) = 3^(24/6) = 3^4 = 81
Answer: x = 81

16. Evaluate : 1 / (log_a bc + 1) + 1 / (log_b ca + 1) + 1 / (log_c ab + 1)
Step 1: 1 = log_a a. So denominator 1 is log_a a + log_a bc = log_a abc.
Step 2: Term 1 = 1 / log_a abc = log_abc a
Step 3: Term 2 = 1 / log_b abc = log_abc b
Step 4: Term 3 = 1 / log_c abc = log_abc c
Step 5: log_abc a + log_abc b + log_abc c = log_abc (abc) = 1
Answer: 1

Case-Study Based Question
Mobile signal strength is measured in decibel-milliwatts (dBm) ... Signal strength S (in dBm) is calculated as: S = 10 log P, where P is the signal power.
(i) Find the signal strength when P = 1 / 10^6.
Step 1: S = 10 log (10^-6) = 10 * (-6) = -60
Answer: -60 dBm
(ii) If signal power is reduced to 1 / 10^10, how does S change?
Step 1: S_new = 10 log (10^-10) = 10 * (-10) = -100 dBm.
Step 2: Change = -100 - (-60) = -40 dBm.
Answer: It decreases by 40 dBm (from -60 dBm to -100 dBm).