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Math Help - Tough logarithmic equation

  1. #1
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    Tough logarithmic equation

    Here's the equation which I have to solve for x,

     \log_5 (5^{\frac{1}{x}} + 125 ) = \log_5 (6) + 1 + \frac{1}{2x}

    I have tried making the RHS into a single logarithm to the base 5 and then taking anti-log......but that didnt help much.

    Please Help.

    Thanks in advance
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  2. #2
    MHF Contributor red_dog's Avatar
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    \log_5\left(5^{\frac{1}{x}}+125\right)=\log_56+ \log_5 \left(5^{1+\frac{1}{2x}}\right)

    \log_5\left(5^{\frac{1}{x}}+125\right)=\log_5\left  (6\cdot 5\cdot 5^{\frac{1}{2x}}\right)

    Then 5^{\frac{1}{x}}+125=30\cdot 5^{\frac{1}{2x}}

    Substitute 5^{\frac{1}{2x}} with y.

    Can you continue?
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  3. #3
    A Plied Mathematician
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    Write the RHS all as log base five quantities. Then the RHS will simplify. Go from there. What do you get?
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