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Thread: Logrithms

  1. #1
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    Logrithms

    This is a problem i'm having trooble on
    i put it as an attachment

    thanks
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  2. #2
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    Hello, anime_mania!


    $\displaystyle \text{(a) If }\left(\log_3x\right)\left(\log_x2x\right)\left(\l og_{2x}y\right) \;=\;\log_xx^2\text{, find the value of }y.$
    On the right side: .$\displaystyle \log_xx^2 \:=\:2\!\cdot\log_xx \:=\:2\cdot1 \:=\:2$


    Use the Base-Change Formula: .$\displaystyle \log_b(x) \:=\:\frac{\log(x)}{\log(b)}$

    Then: .$\displaystyle \log_3x \:=\: \frac{\log x}{\log 3} \qquad \log_x2x \:=\: \frac{\log2x}{\log x} \qquad \log_{2x}y \:=\: \frac{\log y}{\log2x} $


    The equation becomes: .$\displaystyle \frac{\log x}{\log 3}\cdot\frac{\log2x}{\log x}\cdot\frac{\log y}{\log2x} \:=\:2$

    . . which simplifies to: .$\displaystyle \frac{\log y}{\log 3} \:=\:2\quad\Rightarrow\quad \log y \:=\:2\!\cdot\!\log3 \:=\:\log3^2$

    Therefore: .$\displaystyle \log y \:=\:\log9\quad\Rightarrow\quad\boxed{y \:=\:9}$




    $\displaystyle \text{(b) Find integral values of }x,y,z\text{ that satisfy all of the following equations.}$

    . . $\displaystyle {\color{blue}[1]}\;\;z^x \;=\;y^{2x}$
    . . $\displaystyle {\color{blue}[2]}\;\;2^z \;=\;2\cdot4^x$
    . . $\displaystyle {\color{blue}[3]}\;\;x + y + z \;=\;16$

    From [2], we have: .$\displaystyle 2^z \:=\:2\cdot(2^2)^x\:=\:2\cdot2^{2x}\:=\:2^{2x+1}$
    . . Hence: .$\displaystyle 2x + 1 \:=\:z\quad\Rightarrow\quad x \:=\:\frac{z-1}{2}$ .[4]

    From [1], we have: .$\displaystyle z^x \:=\:(y^2)^x\quad\Rightarrow\quad z = y^2$ .[5]

    Substitute [5] into [4]: .$\displaystyle x \:=\:\frac{y^2-1}{2}$ .[6]

    Substitute [5] and [6] into [3]: .$\displaystyle \frac{y^2-1}{2} + y + y^2 \:=\:16$

    . . which simplifies to the quadratic: .$\displaystyle 3y^2 + 2y - 33 \:=\:0$

    . . which factors: .$\displaystyle (y - 3)(3y + 11) \:=\:0$

    . . and has the integral solution: .$\displaystyle \boxed{y \:=\:3}$

    Substitute into [6]: .$\displaystyle x \:=\:\frac{3^2-1}{2}\quad\Rightarrow\quad\boxed{x \:=\:4}$

    Substitute into [5]: .$\displaystyle z \:=\:3^2\quad\Rightarrow\quad\boxed{z \:=\:9}$

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