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Thread: Simpel Log

  1. #1
    Junior Member
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    Wink Simpel Log

    log3(x+1)=log9(26+2x)
    X?


    with calculation...
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  2. #2
    MHF Contributor Amer's Avatar
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    Quote Originally Posted by sf1903 View Post
    log3(x+1)=log9(26+2x)
    X?


    with calculation...

    $\displaystyle \log _{3} (x+1) = \log _{9} (26+2x) $

    $\displaystyle 26+2x = 9^{\log_3 (x+1)}$

    $\displaystyle 26+2x = (3^{\log_3 (x+1)})^2 $

    $\displaystyle 26+2x = (x+1)^2 $ you can continue from here
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  3. #3
    Senior Member pacman's Avatar
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    we may use this logarithmic identity by EULER, log (base b) a = (log a)/(log b)

    solve for x: log3(x+1)=log9(26+2x)

    -------------------------------------------------------

    log (x + 1)/(log 3) = log (2x + 26)/(log 9) = log (2x + 26)/(2 log 3)

    cross-multiply,

    2 (log 3) log (x + 1) = (log 3) log (2x + 26), cancel log 3

    2 log (x + 1) = log (2x + 26),

    log (x + 1)^2 = log (2x + 26),

    take anti-log og both sides,

    (x + 1)^2 = 2x + 26,

    x^2 + 2x + 1 = 2x + 26,

    2x^2 - 2x + 2x - 26 +1 = 0,

    x^2 - 25 = 0

    factoring,

    (x + 5)(x - 5) =0,

    x = {-5, 5}, but only 5 is the valid answer

    the graph below is wrong
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  4. #4
    Senior Member pacman's Avatar
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    x^2 - 25 = 0, this is the graph
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  5. #5
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    Hello, sf1903!

    Another approach . . .


    Solve for $\displaystyle x\!:\;\;\log_3(x+1)\:=\:\log_9(2x+26)$

    Let: .$\displaystyle \log_3(x+1) \:=\:P \quad\Rightarrow\quad 3^P \:=\:x+1 $

    Square both sides: .$\displaystyle (3^P)^2 \:=\:(x+1)^2 \quad\Rightarrow\quad (3^2)^P \:=\:(x+1)^2 \quad\Rightarrow\quad 9^P \:=\:(x+1)^2 \quad\Rightarrow\quad P \:=\:\log_9(x+1)^2$

    . . We have: .$\displaystyle \log_3(x+1) \:=\:\log_9(x+1)^2$


    The equation becomes: .$\displaystyle \log_9(x+1)^2 \:=\:\log_9(2x+26)$

    Exponentiate both sides: .$\displaystyle (x+1)^2 \:=\:2x + 26$

    . . and so on . . .

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