EVALUATE LOGARITHMS WITHOUT USING CALCULATOR

Find :

Problem 1 :

log₃ 9

Solution :

= log₃ 9

= log₃ 3²

= 2 log₃ 3

= 2 (1)

= 2

Problem 2 :

log₂ 32

Solution :

= log₂ 32

= log₂ 2⁵

= 5 log₂ 2

= 5 (1)

= 5

Problem 3 :

log₂ √2

Solution :

= log₂ √2

= log₂ 2^1/2

= 1/2 log₂ 2

= 1/2 (1)

= 1/2

Problem 4 :

log₄ √2

Solution :

= log₄ √2

Problem 5 :

log₃ 3√3

Solution :

= log₃ 3√3

Problem 6 :

log₆ 1

Solution :

= log₆ 1

= 0

Problem 7 :

log₈ 8

Solution :

= log₈ 8

= 1

Problem 8 :

log₈ (1/8)

Solution :

= log₈ (1/8)

= log₈ 8^-1

= (-1) log₈ 8

= -1 × 1

= -1

Problem 9 :

log_1/8 (1/8)

Solution :

= log_1/8 (1/8)

= 1

Problem 10 :

log_√2 (1/√2)

Solution :

= log_√2 (1/√2)

= log_√2 √2^-1

= (-1) log_√2 √2

= -1 × 1

= -1

Problem 11 :

log₂ (1/√2)

Solution :

= log₂ (1/√2)

= log₂ 2^-1/2

= (-1/2) log₂ 2

= -1/2 × 1

= -1/2

Problem 12 :

log₈ (1/√2)

Solution :

= log₈ (1/√2)

= log₈ (1/2)^1/2

= log₈ (2)^-1/2

Problem 13 :

Solve ln (4x − 7) = ln (x + 5)

Solution :

ln (4x − 7) = ln (x + 5)

Cancelling ln on both sides

4x - 7 = x + 5

4x - x = 5 + 7

3x = 12

x = 12/3

x = 4

Problem 14 :

log₂(5x − 17) = 3

Solution :

log₂(5x − 17) = 3

Moving logarithm to the other side of the equal sign.

5x - 17 = 2³

5x - 17 = 8

5x = 8 + 17

5x = 25

x = 25/5

x = 5

So, the value of x is 5.

Problem 15 :

log₁₀(9x + 1) = 3

Solution :

log₁₀(9x + 1) = 3

9x + 1 = 10³

9x + 1 = 1000

9x = 1000 - 1

9x = 999

x = 999/9

x = 111

Problem 16 :

log₈√(1 - x) = 1/3

Solution :

log₈√(1 - x) = 1/3

√(1 - x) = 8^1/3

√(1 - x) = (2³)^1/3

√(1 - x) = 2^{3 x (1/3)}

√(1 - x) = 2

1 - x = 2²

1 - x = 4

x = 1 - 4

x = -3

So, the value of x is -3.

Problem 17 :

Solve the simultaneous equations:

log₃ x + log₅ y = 6

log₃ x - log₅ y = 2

Solution :

log₃ x + log₅ y = 6 -----(1)

log₃ x - log₅ y = 2 -----(2)

(1) + (2)

2log₃ x = 6 - 2

2log₃ x = 4

log₃ x = 4/2

log₃ x = 2

x = 3²

x = 9

Applying the value of x in (1), we get

log₃ 9 + log₅ y = 6

log₃ 3² + log₅ y = 6

2 log₃ 3 + log₅ y = 6

2 (1) + log₅ y = 6

2 + log₅ y = 6

log₅ y = 6 - 2

log₅ y = 4

y = 5⁴

y = 625

So, the values of x and y are 9 and 625 respectively.

Problem 18 :

Solve the equation

3(1 + log x) = 6 + log x

Solution :

3(1 + log x) = 6 + log x

3 + 3 log x = 6 + log x

3 log x - log x = 6 - 3

2 log x = 3

log x = 3/2

log x = 1.5

x = 10^1.5

x = 31.6

Problem 19 :

Solve log₃ [(2 + x) / (2 - x)] = 3

Solution :

log₃ [(2 + x) / (2 - x)] = 3

[(2 + x) / (2 - x)] = 3³

[(2 + x) / (2 - x)] = 27

2 + x = 27(2 - x)

2 + x = 54 - 2x

x + 2x = 54 - 2

3x = 56

x = 56/3

So, the value of x is 56/3.