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