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Math Help - what rule?

  1. #1
    Newbie yuriythebest's Avatar
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    what rule?

    can you please explain how this occurs:

    [IMG]file:///E:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
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  2. #2
    Super Member flyingsquirrel's Avatar
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    Hi

    If \ln(\sqrt{x},\,10) is \ln_{\sqrt{x}}10 then simply use the definition of \ln_ab=\frac{\ln b}{\ln a} :

    \ln_{\sqrt{x}}10=\frac{\ln 10}{\ln\sqrt{x}}=\frac{\ln 10}{\frac{1}{2}\ln x}=2\underbrace{\frac{\ln10}{\ln x}}_{\ln_x10}=2\ln_x10

    Hope that helps.
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by yuriythebest View Post
    can you please explain how this occurs:

    [IMG]file:///E:/DOCUME%7E1/ADMINI%7E1/LOCALS%7E1/Temp/moz-screenshot.jpg[/IMG]
    What machine and what application is this from?

    RonL
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  4. #4
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    Hello, yuriythebest!

    Show that: . \log_{\sqrt{x}}(10) \;=\;2\cdot\log_x(10)
    I constructed a general proof of this years ago . . .


    \text{Consider:}\:\log_b(N) \:=\:p \;\;{\color{blue}[1]}

    Then: . b^p \:=\:N

    Square both sides: . \left(b^p\right)^2 \:=\:N^2 \quad\Rightarrow\quad (b^2)^p \:=\:N^2

    Take logs (base b^2): . \log_{b^2}(b^2)^p \:=\: \log_{b^2}(10^2)\quad\Rightarrow\quad p\cdot\log_{b^2}(b^2) \:=\:2\cdot\log_{b^2}(10^2)

    Since \log_{b^2}(b^2) = 1, we have: . p \:=\:2\cdot\log_x(10)\;\;{\color{blue}[2]}

    Equate [1] and [2]: . \boxed{\log_b(10) \;=\;2\cdot\log_{b^2}(10)}


    In baby-talk: "If we square the base, double the log."


    And this can be generalized: "Cube the base, triple the log", etc.
    .
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  5. #5
    Newbie yuriythebest's Avatar
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    thanks everyone!

    captainBlack- this was from the microsoft student graphing calculator. very user friendly.
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