SOLUTION: Find x if \log_2 x^2 + \log_{1/2} x + 3 \log_4 x = 5.

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Question 1209836: Find x if \log_2 x^2 + \log_{1/2} x + 3 \log_4 x = 5.
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
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Find x if log_2 x^2 + log_{1/2} x + 3*log_4 x = 5.
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      Step by step


The domain to this equation is the set of all real positive numbers,
so we work in this domain.


    log%282%2Cx%5E2%29 = 2%2Alog%282%2Cx%29%29.


    log%281%2F2%2Cx%29 = -log%282%2Cx%29.


    3%2Alog%284%2Cx%29 = %283%2F2%29%2Alog%282%2Cx%29.


Therefore, the whole given equation is equivalent to this one

    2%2Alog%282%2Cx%29%29 - log%282%2Cx%29 + %283%2F2%29%2Alog%282%2Cx%29 = 5.


Simplify

    %285%2F2%29%2Alog%282%2Cx%29 = 5,

    log%282%2Cx%29 = 5%2F%28%285%2F2%29%29 = 2

     x = 2%5E2 = 4.


ANSWER.  x = 4.

Solved.