Solve: log 2  (3x )  - log 4 ( x-2) = 3

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

We know that log a b = log c b/ log c a

==> log 4 (x-2) = log 2 (x-2)/ log 2 (4) = log 2 (x-2)/2 = (1/2) log (x-2) = log (x-2)^1/2

==> loh 2 (3x) - log (x-2)^1/2 = 3

We know that log a - log b = log a/b

==> loh 2 (3x)/(x-2)^1/2 = 3

==> 3x/ (x-2)^1/2  = 2^3

==>  3x = 8*(x-2)^1/2

Now square both sides:

==> 9x^2 = 64(x-2)

==> 9x^2 = 64x - 128

==> 9x^2 - 64x + 128 = 0

==> x1= [64 + sqrt(4096 - 4608)/18     ( impossible

Then the problem has no real solution.

 

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