PROBLEMS ON SOLVING LOGARITHMIC EQUATIONS

Solve the following logarithmic equations.

Problem 1 :

ln x = -3

Solution:

ln x = -3

x = e^-3

Problem 2 :

log(3x - 2) = 2

Solution:

log(3x - 2) = 2

log₁₀(3x - 2) = 2

3x - 2 = 10²

3x - 2 = 100

3x = 102

x = 102/3

x = 34

Problem 3 :

2 logx = log2 + log(3x - 4)

Solution:

2 log x = log 2 + log(3x - 4)

x² = 2(3x - 4)

x² = 6x - 8

x² - 6x + 8 = 0

(x - 4)(x - 2) = 0

x = 4, x = 2

Problem 4 :

log x + log(x - 1) = log(4x)

Solution:

log x + log(x - 1) = log(4x)

x(x - 1) = 4x

x² - x = 4x

x² - 4x - x = 0

x² - 5x = 0

x(x - 5) = 0

x = 0, x = 5

Problem 5 :

log₃(x + 25) - log₃(x - 1) = 3

Solution:

log₃(x + 25) - log₃(x - 1) = 3

Problem 6 :

log₉(x - 5) + log₉(x + 3) = 1

Solution:

log₉(x - 5) + log₉(x + 3) = 1

log₉ (x - 5) (x + 3) = log₉ 9 

(x - 5)(x + 3) = 9

x² - 2x - 24 = 0

(x - 6)(x + 4) = 0

x = 6, x = -4

Problem 7 :

log x + log(x - 3) = 1

Solution:

log x + log(x - 3) = 1

log₁₀ x + log₁₀(x - 3) = 1

log₁₀(x(x - 3)) = 1

log₁₀(x² - 3x) = 1

x² - 3x = 10¹

x² - 3x - 10 = 0

(x - 5) (x + 2) = 0

x = 5 or x = -2

Problem 8 :

log₂(x - 2) + log₂(x + 1) = 2

Solution:

log₂(x - 2) + log₂(x + 1) = 2

log₂((x - 2)(x + 1)) = 2

log₂(x² - x - 2) = 2

x² - x - 2 = 2²

x² - x - 2 = 4

x² - x - 6 = 0

(x - 3)(x + 2) = 0

x = 3 or x = -2

Problem 9 :

Given that 

2 log₃(x - 5) - log₃(2x - 13) = 1

Show that x² - 16x + 64 = 0 and solve for x.

Solution :

2 log₃(x - 5) - log₃(2x - 13) = 1

log₃(x - 5)² - log₃(2x - 13) = 1

log₃[(x - 5)² / (2x - 13)] = 1

[(x - 5)² / (2x - 13)] = 3¹

[(x - 5)² / (2x - 13)] = 3

(x - 5)² = 3(2x - 13)

x² - 10x + 25 = 6x - 39

x² - 10x - 6x + 25 + 39 = 0

x² - 16x + 64 = 0

x² - 8 x - 8x + 64 = 0

x(x - 8) - 8(x - 8) = 0

(x - 8)(x - 8) = 0

x = 8 and x = 8

Problem 10 :

a) Find the positive value of x such that 

log _x64 = 2

b) Solve for x 

log₂(11 - 6x) = 2log₂(x - 1) + 3

Solution :

a) 

log _x64 = 2

64 = x²

8² = x²

Equating the bases, we get x = 8

b) 

log₂(11 - 6x) = 2log₂(x - 1) + 3

log₂(11 - 6x) - 2log₂(x - 1) = 3

log₂(11 - 6x) - log₂(x - 1)² = 3

log₂[(11 - 6x)/(x - 1)²] = 3

[(11 - 6x)/(x - 1)²] = 2³

11 - 6x = 8(x - 1)²

11 - 6x = 8(x² - 2x + 1)

11 - 6x = 8x² - 16x + 8

8x² - 16x + 6x + 8 - 11 = 0

8x² - 10x - 3 = 0

8x² - 12x + 2x - 3 = 0

4x(2x - 3) + 1(2x - 3) = 0

(4x + 1)(2x - 3) = 0

x = -1/4 and x = 3/2

Problem 11 :

Given that a and b are positive constants, solve the simultaneous equations

a = 3b

log₃ a + log₃ b = 2

Give your answers as exact numbers.

Solution :

a = 3b -------(1)

log₃ a + log₃ b = 2 -------(2)

Applying (1) in (2)

log₃ 3b + log₃ b = 2

log₃ (3b ⋅ b) = 2

log₃ (3b²) = 2

3b² = 3²

3b² = 9

b² = 9/3

b² = 3

b = √3

b = √3 and -√3

Applying the value of b, we get

a = 3√3 and a = -3√3