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Math Help - Logarithm question

  1. #1
    Junior Member
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    Logarithm question

    Here is the question:

    Find: (logbase2 (x))^2

    if logbase2 (logbase8 (x)) = logbase8 (logbase2 (x))

    If you could go slow on me, that would be great.
    Any help is appreciated.
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  2. #2
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    Hello, Shapeshift!

    Given: . \log_2\left(\log_8X\right) \:=\:\log_8\left(\log_2X\right)

    Find: . (\log_2X)^2

    Let: \log_8Y \:=\:p \quad\Rightarrow\quad 8^p \:=\:Y \quad\Rightarrow\quad \left(2^3\right)^p \:=\:Y \quad\Rightarrow\quad 2^{3p} \:=\:Y

    Take logs (base 2): . \log_2\left(2^{3p}\right) \:=\:\log_2Y \quad\Rightarrow\quad 3p\log_22 \:=\:\log_2Y

    Since \log_22 = 1, we have: . 3p \:=\:\log_2Y \quad\Rightarrow\quad p \:=\:\tfrac{1}{3}\log_2Y

    . . Hence: . \log_8Y \:=\:\tfrac{1}{3}\log_2Y



    \text{The equation: }\;\log_2\underbrace{\left(\log_8X\right)} \;\;=\;\;\underbrace{\log_8\left(\log_2X\right)}
    . . \text{becomes: }\;\log_2\overbrace{\left(\tfrac{1}{3}\log_2X\righ  t)} \;=\;\overbrace{\tfrac{1}{3}\log_2\left(\log_2X\ri  ght)}


    Multiply by 3: . 3\log_2\left(\tfrac{1}{3}\log_2X\right) \;=\;\log_2\left(\log_2X\right) \quad\Rightarrow\quad \log_2\left(\tfrac{1}{3}\log_2X\right)^3 \;=\;\log_2\left(\log_2X\right)

    Exponentiate both sides: . \left(\tfrac{1}{3}\log_2X\right)^3 \;=\;\log_2X \quad\Rightarrow\quad \tfrac{1}{27}\left(\log_2X\right)^3 \;=\;\log_2X

    Multiply by 27: . \left(\log_2X\right)^3 \;=\;27\log_2X \quad\Rightarrow\quad \left(\log_2X\right)^3 - \log_2X\;=\;0

    Factor: . \log_2X\bigg[(\log_2X)^2 - 27\bigg] \;=\;0


    We have two equations to solve . . .

    . . (1)\;\;\log_2X \:=\:0\quad\hdots] .
    which is not allowed in the original equation

    . . [2]\;\;\left(\log_2X\right)^2 - 27 \:=\:0 \quad\Rightarrow\quad {\color{blue}\left(\log_2X\right)^2 \:=\:27}

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