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

(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

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