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
log_{10}(3x - 2) = 2
3x - 2 = 10^{2}
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)
x^{2} = 2(3x - 4)
x^{2} = 6x - 8
x^{2} - 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
x^{2} - x = 4x
x^{2} - 4x - x = 0
x^{2} - 5x = 0
x(x - 5) = 0
x = 0, x = 5
Problem 5 :
log_{3}(x + 25) - log_{3}(x - 1) = 3
Solution:
log_{3}(x + 25) - log_{3}(x - 1) = 3
Problem 6 :
log_{9}(x - 5) + log_{9}(x + 3) = 1
Solution:
log_{9}(x - 5) + log_{9}(x + 3) = 1
log_{9} (x - 5) (x + 3) = log_{9} 9
(x - 5)(x + 3) = 9
x^{2} - 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
log_{10} x + log_{10}(x - 3) = 1
log_{10}(x(x - 3)) = 1
log_{10}(x^{2} - 3x) = 1
x^{2} - 3x = 10^{1}
x^{2} - 3x - 10 = 0
(x - 5) (x + 2) = 0
x = 5 or x = -2
Problem 8 :
log_{2}(x - 2) + log_{2}(x + 1) = 2
Solution:
log_{2}(x - 2) + log_{2}(x + 1) = 2
log_{2}((x - 2)(x + 1)) = 2
log_{2}(x^{2} - x - 2) = 2
x^{2} - x - 2 = 2^{2}
x^{2} - x - 2 = 4
x^{2} - x - 6 = 0
(x - 3)(x + 2) = 0
x = 3 or x = -2
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