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Thread: Solve the logarithmic equation

  1. #1
    Senior Member DeMath's Avatar
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    Solve the logarithmic equation

    Solve analytically the logarithmic equation

    $\displaystyle \log_{x+1}(\sqrt{2}-1)\cdot\log_{4-2\sqrt{2}}(x^2+2x+2)=1$
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  2. #2
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    Let $\displaystyle a=\sqrt{2}-1$ and $\displaystyle b=x+1$. Then the equation becomes:

    $\displaystyle \displaystyle{\frac{\ln{a}}{\ln{b}}\cdot \frac{\ln(b^2+1)}{\ln(a^2+1)}}=1$

    or equivalently:

    $\displaystyle \displaystyle{\frac{\ln{a}}{\ln(a^2+1)}=\frac{\ln b}{\ln(b^2+1)}}$

    Now, the derivative of $\displaystyle \frac{\ln{t}}{\ln(t^2+1)}$ is

    $\displaystyle \frac{\frac{\ln(t^2+1)}{t}-\frac{2t\ln t}{t^2+1}}{\ln^2(t^2+1)} = \frac{(t^2+1)\ln(t^2+1)-t^2\ln t^2}{t(t^2+1)\ln^2(t^2+1)} > 0$

    So we conclude that the only solution is $\displaystyle a=b$, or:

    $\displaystyle x=\sqrt{2}-2$
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