Results 1 to 2 of 2

Math Help - Solve the logarithmic equation

  1. #1
    Senior Member DeMath's Avatar
    Joined
    Nov 2008
    From
    Moscow
    Posts
    473
    Thanks
    5

    Solve the logarithmic equation

    Solve analytically the logarithmic equation

    \log_{x+1}(\sqrt{2}-1)\cdot\log_{4-2\sqrt{2}}(x^2+2x+2)=1
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Member
    Joined
    Nov 2009
    Posts
    106
    Let a=\sqrt{2}-1 and b=x+1. Then the equation becomes:

    \displaystyle{\frac{\ln{a}}{\ln{b}}\cdot \frac{\ln(b^2+1)}{\ln(a^2+1)}}=1

    or equivalently:

    \displaystyle{\frac{\ln{a}}{\ln(a^2+1)}=\frac{\ln b}{\ln(b^2+1)}}

    Now, the derivative of \frac{\ln{t}}{\ln(t^2+1)} is

    \frac{\frac{\ln(t^2+1)}{t}-\frac{2t\ln t}{t^2+1}}{\ln^2(t^2+1)} = \frac{(t^2+1)\ln(t^2+1)-t^2\ln t^2}{t(t^2+1)\ln^2(t^2+1)} > 0

    So we conclude that the only solution is a=b, or:

    x=\sqrt{2}-2
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. solve logarithmic equation
    Posted in the Algebra Forum
    Replies: 3
    Last Post: August 28th 2009, 10:45 AM
  2. solve the logarithmic equation
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: March 31st 2008, 10:37 AM
  3. solve logarithmic equation
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: March 31st 2008, 05:58 AM
  4. solve the logarithmic equation
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: March 31st 2008, 05:46 AM
  5. Please solve logarithmic equation
    Posted in the Pre-Calculus Forum
    Replies: 2
    Last Post: November 15th 2007, 05:59 AM

Search Tags


/mathhelpforum @mathhelpforum