Results 1 to 2 of 2

Math Help - Logarithmic Equation

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    11

    Question Logarithmic Equation

    Let r = \log_{b} \frac{8}{45} and s = \log_{b} \frac{135}{4}.

    Find an ordered pair of integers (m,n) such that \log_{b} \frac{32}{5} = mr+ns.

    Can someone help me set this up? I don't see how to solve for 2 variables with only one equation.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Grandad's Avatar
    Joined
    Dec 2008
    From
    South Coast of England
    Posts
    2,570
    Thanks
    1
    Hello fizz
    Quote Originally Posted by fizz View Post
    Let r = \log_{b} \frac{8}{45} and s = \log_{b} \frac{135}{4}.

    Find an ordered pair of integers (m,n) such that \log_{b} \frac{32}{5} = mr+ns.

    Can someone help me set this up? I don't see how to solve for 2 variables with only one equation.
    \log_{b} \frac{32}{5} = mr+ns
    =m\log_{b} \frac{8}{45}+n\log_{b} \frac{135}{4}

    =\log_{b}\left( \frac{8}{45}\right)^m+\log_{b}\left( \frac{135}{4}\right)^n

    =\log_{b}\left( \frac{8}{45}\right)^m\left( \frac{135}{4}\right)^n
    \Rightarrow \frac{32}{5}=\frac{8^m135^n}{45^m4^n}

    Now express all these numbers in their prime factors:
    32=2^5,\quad 8=2^3,\quad 135 = 3^3\times5,\quad 45 =  3^2\times5,\quad 4=2^2
    \Rightarrow \frac{2^5}{5^1}=2^5\,5^{-1}=\frac{2^{3m}\,3^{3n}\,5^n}{3^{2m}\,5^m\,2^{2n}}  =2^{3m-2n}\,3^{3n-2m}\,5^{n-m}

    So if we now compare the indices, we get:

    2: 5 = 3m - 2n

    5: -1 = n-m

    \Rightarrow m=3, n=2

    Check the power of 3: 0=3n-2m, which is correct.

    So the ordered pair is (3,2).

    Grandad
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Logarithmic equation
    Posted in the Algebra Forum
    Replies: 2
    Last Post: October 15th 2010, 08:14 AM
  2. Logarithmic equation
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: August 20th 2010, 11:07 PM
  3. Logarithmic Equation
    Posted in the Algebra Forum
    Replies: 3
    Last Post: September 27th 2009, 01:41 AM
  4. logarithmic equation
    Posted in the Algebra Forum
    Replies: 1
    Last Post: May 13th 2009, 12:55 PM
  5. Logarithmic Equation
    Posted in the Algebra Forum
    Replies: 1
    Last Post: April 21st 2009, 02:07 PM

Search Tags


/mathhelpforum @mathhelpforum