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Math Help - Logarithms and patterns

  1. #1
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    Smile Logarithms and patterns

    Here is a quetion I couldn't solve.

    Let loga (x) = c and logb (x) = d.
    Find the general statement that expresses logab (x), in terms of c and d.

    I would appreciate your help, thanks a lot,
    Amine
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  2. #2
    Junior Member rubix's Avatar
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    loga (x) = c

    logb (x) = d

    logab (x) = ?


    a^c = x and b^d = x

    a = x ^ (1/c) and b = x ^ (1/d)

    ab = x^ 1/c x ^ 1/d

    = x ^ (d+c/cd)

    logab (x) = d+c / cd


    humm i might have flipped something
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  3. #3
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    Hello, rubix!

    Ya done good!

    I'll put it in LaTex . . .


    \begin{array}{c}\log_a (x) \:=\: c \\ \log_b(x) \:=\:d\end{array}\qquad \log_{ab}(x) \:=\: ?

    \begin{array}{ccccc}a^c \:= \:x & \Rightarrow & a \:=\:x^{\frac{1}{x}} & [1] \\ b^d \:=\: x & \Rightarrow & b \:=\:x^{\frac{1}{d}} & [2]\end{array}


    Multiply [1] and [2]: . ab \:=\:x^{\frac{1}{c}}\cdot x^{\frac{1}{d}} \;=\;x^{\frac{d+c}{cd}}


    Therefore: . \log_{ab}(x) \:=\: \frac{c+d}{cd}


    Good work!

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  4. #4
    Junior Member rubix's Avatar
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    thnx for conforming Soroban...and thnx for nice well formatted answe.

    cheers.
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  5. #5
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    Thanks a lot for your help!
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