SOLVING LOGARITHMIC EQUATONS USING PROPERTIES

To solve a logarithmic equation, first combine the logarithmic terms using rules of logarithms.

Some of the rules in logarithm :

log m + log n = log (m x n)

log m - log n = log (m / n)

log mⁿ = n log m

log _a a = 1

After combining more than one logarithmic terms as one term, we have to convert logarithmic form to exponential form then solve for the variable.

Conversion from logarithmic form to exponential form :

Solve the following logarithmic equations :

Problem 1 :

log₇ 3 + log₇ x = log₇ 32

Solution :

log₇ 3 + log₇ x = log₇ 32

log_a m + log_a n = log_a mn

log₇ (3x) = log₇ 32

3x = 32

x = 32/3

Problem 2 :

2 log₆ 4x = 0

Solution :

2 log₆ 4x = 0

log₆ 4x = 0

4x = 6⁰

4x = 1

x = 1/4

Problem 3 :

log₂ x + log₂ (x - 3) = 2

Solution :

log₂ x + log₂ (x - 3) = 2

log₂ (x) (x - 3) = 2

log₂ (x² - 3x) = 2

x² - 3x = 2²

x² - 3x = 4

x² - 3x - 4 = 0

x² - 4x + x - 4 = 0

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

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

x = 4 or x = -1

Problem 4 :

log₂ (x + 5) - log₂ (x - 2) = 3

Solution :

log₂ (x + 5) - log₂ (x - 2) = 3

log₂ (x + 5 / x - 2) = 3

log₂ (x + 5 / x - 2) = log₂ 2³

(x + 5) / (x - 2) = 8

x + 5 = 8(x - 2)

x + 5 = 8x - 16

8x - x = 16 + 5

7x = 21

x = 3

Problem 5 :

4 ln (2x + 3) = 11

Solution :

4 ln (2x + 3) = 11

ln (2x + 3) = 11/4

ln (2x + 3) = 2.75

2x + 3 = e^2.75

2x + 3 = 15.6426

2x = 15.6426 - 3

2x = 12.6426

x = 12.6426/2

x = 6.321

Problem 6 :

log x - log 6 = 2 log 4

Solution :

log x - log 6 = 2 log 4

log (x/6) = log 4²

x/6 = 4²

x/6 = 16

x = 16 × 6

x = 96

Problem 7 :

log 2x = 1.5

Solution :

log 2x = 1.5

2x = 10^1.5

x = 10^1.5/2

x = 31.62/2

x = 15.81

Problem 8 :

log₂ 2x = -0.65

Solution :

log₂ 2x = -0.65

2x = 2^-0.65

2x = 0.637

x = 0.637/2

x = 0.32

Problem 9 :

1/3 log₂ x + 5 = 7

Solution :

1/3 log₂ x + 5 = 7

1/3 log₂ x = 7 - 5

1/3 log₂ x = 2

log₂ x = 6

x = 2⁶

Problem 10 :

4 log₅ (x + 1) = 4.8

Solution :

4 log₅ (x + 1) = 4.8

log₅ (x + 1) = 4.8/4

log₅ (x + 1) = 1.2

x + 1 = 5^1.2

x = 5^1.2 - 1

x = 6.90 - 1

x = 5.90

Problem 11 :

log₂ x + log₂ 3 = 3

Solution :

log₂ x + log₂ 3 = 3

log₂ (3x) = 3

3x = 2³

3x = 8

x = 8/3

Problem 12 :

2 log₄ x - log₄ (x - 1) = 1

Solution :

2 log₄ x - log₄ (x - 1) = 1

log₄ x² - log₄ (x - 1) = 1

log₄ (x²/x - 1) = 1

x²/x - 1 = 4

x² = 4(x - 1)

x² = 4x - 4

x² - 4x + 4 = 0

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

x = 2