Results 1 to 2 of 2

Math Help - System of Equations - Logarithms

  1. #1
    Banned
    Joined
    Sep 2008
    Posts
    47

    System of Equations - Logarithms

    Solve the following system of equations in positive real numbers (the logarithm is in base 10):

    \log (2xy)=\log x \log y
    \log(yz)=\log y \log z
    \log(2zx)=\log z^2 \log x
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Opalg's Avatar
    Joined
    Aug 2007
    From
    Leeds, UK
    Posts
    4,041
    Thanks
    7
    Quote Originally Posted by Winding Function View Post
    Solve the following system of equations in positive real numbers (the logarithm is in base 10):

    \log (2xy)=\log x \log y
    \log(yz)=\log y \log z
    \log(2zx)=\log z^2 \log x
    Let a=\log x,\;b=\log y,\;c=\log z,\;m=\log 2. Then the equations become

    m+a+b=ab,\quad b+c=bc,\quad m+c+a = 2ca.

    The first two equations can be written (a-1)(b-1)=m+1 and (b-1)(c-1)=1, from which it follows that a-1 = (m+1)(c-1) and so a = (m+1)c-m. Substitute that into the third equation, and you find that c = \frac{3m+2}{2m+2}, from which it follows that a = \frac{m+2}2 and b = \frac{3m+2}m.

    So 2a = m+2, or 2\log x = \log 2 + \log 100, from which x^2 = 200, x = 10\sqrt2. The expressions for y and z are similar but more awkward, because they will involve raising numbers to irrational powers.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Solve an equation system with logarithms
    Posted in the Algebra Forum
    Replies: 2
    Last Post: October 7th 2011, 09:16 AM
  2. Exponential Equations with Logarithms
    Posted in the Algebra Forum
    Replies: 2
    Last Post: August 10th 2010, 08:55 AM
  3. Replies: 2
    Last Post: April 20th 2010, 03:26 PM
  4. logarithms and domain, logarithm equations
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: April 14th 2009, 05:12 AM
  5. Replies: 2
    Last Post: November 21st 2008, 09:57 PM

Search Tags


/mathhelpforum @mathhelpforum