Results 1 to 5 of 5

Math Help - Logarithmic Proof

  1. #1
    Senior Member Stroodle's Avatar
    Joined
    Jun 2009
    Posts
    367

    Logarithmic Proof

    Given 3^x=4^y=12^z show that z=\frac{xy}{x+y}

    Thanks for you help!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,564
    Thanks
    1425
    Quote Originally Posted by Stroodle View Post
    Given 3^x=4^y=12^z show that z=\frac{xy}{x+y}

    Thanks for you help!
    What a horrid question...


    First note that 12^z = (3 \cdot 4)^z = 3^z \cdot 4^z

    Focus firstly on

    3^x = 12^z

    3^x = 3^z \cdot 4^z

    3^{x - z} = 4^z

    \log{3^{x - z}} = \log{4^z}

    (x - z)\log{3} = z\log{4}

    \frac{x - z}{z} = \frac{\log{4}}{\log{3}}.


    Now focus on

    4^y = 3^z \cdot 4^z

    4^{y - z} = 3^z

    \log{4^{y - z}} = \log{3^z}

    (y - z)\log{4} = z\log{3}

    \frac{\log{4}}{\log{3}} = \frac{z}{y - z}.



    Therefore

    \frac{x - z}{z} = \frac{z}{y - z}

    \frac{x}{z} - 1 = \frac{y}{y - z} - 1

    \frac{x}{z} = \frac{y}{y - z}

    \frac{y - z}{z} = \frac{y}{x}

    \frac{y}{z} - 1 = \frac{y}{x}

    \frac{y}{z} = \frac{y}{x} + 1

    \frac{y}{z} = \frac{x + y}{x}

    \frac{z}{y} = \frac{x}{x + y}

    z = \frac{xy}{x + y}.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Senior Member Stroodle's Avatar
    Joined
    Jun 2009
    Posts
    367
    Awesome. I get it now.

    Thanks for your help Prove It.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello Stroodle
    Quote Originally Posted by Stroodle View Post
    Given 3^x=4^y=12^z show that z=\frac{xy}{x+y}

    Thanks for you help!
    Here's an alternative approach:

    3^x = 12^z

    \Rightarrow x\log3 = z\log12

    \Rightarrow x = \frac{z\log12}{\log 3}

    Similarly y = \frac{z\log12}{\log4}

    \Rightarrow \frac{xy}{x+y}= \frac{\dfrac{z\log12}{\log 3}\cdot\dfrac{z\log12}{\log 4}}{\dfrac{z\log12}{\log 3}+\dfrac{z\log12}{\log 4}}

    = \frac{\dfrac{z\log12}{\log 3}\cdot\dfrac{z\log12}{\log 4}}{\dfrac{z\log12}{\log 3}+\dfrac{z\log12}{\log 4}}\color{red}\times\frac{\log3\log4}{\log3\log4}

    = \frac{z^2(\log12)^2}{z\log12(\log4+\log3)}

    =\frac{z\log12}{\log12}

    =z

    Grandad
    Follow Math Help Forum on Facebook and Google+

  5. #5
    MHF Contributor
    Joined
    Sep 2008
    From
    West Malaysia
    Posts
    1,261
    Quote Originally Posted by Stroodle View Post
    Given 3^x=4^y=12^z show that z=\frac{xy}{x+y}

    Thanks for you help!
    HI

    Or another approach .

    3^x=4^y=12^z=k

    3^x=k\Rightarrow 3=k^{\frac{1}{x}}

    4^y=k\Rightarrow 4=k^{\frac{1}{y}}

    12^z=k\Rightarrow 12=k^{\frac{1}{z}}

    we know for the fact that 4 x 3 = 12

    so k^{\frac{1}{x}}\times k^{\frac{1}{y}}=k^{\frac{1}{z}}

     \frac{1}{x}+\frac{1}{y}=\frac{1}{z}

    yz+xz=xy

    z(x+y)=xy

    z=xy/(x+y)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Need Help Solving a Proof for a logarithmic function
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: January 30th 2010, 12:02 AM
  2. More logarithmic help
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 8th 2009, 06:14 AM
  3. logarithmic
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 24th 2008, 10:02 PM
  4. Logarithmic Help
    Posted in the Algebra Forum
    Replies: 8
    Last Post: July 18th 2008, 01:49 AM
  5. logarithmic
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: November 15th 2006, 03:56 AM

Search Tags


/mathhelpforum @mathhelpforum