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Math Help - Exponential Contest Problem

  1. #1
    Junior Member
    Joined
    Jul 2008
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    74

    Exponential Contest Problem

    Hello everyone,

    Could someone pleae check my solution for the following problem?

    Thank you!

    ---

    1. Let  a and  b be real numbers with  a > 1 and  b > 0 . If  ab = a^b and  \frac{a}{b} = a^{3b} , determine the value of  a  .

    ---

    When the first equation is multiplied by a:

     b = a^{b - 1} --(3)

    After substitution of (3) in the second equation:

     \frac{a}{a^{b - 1}} = a^{3b}

     a^{2 - b} = a^{3b}

    Therefore:

     2 - b = 3b \Rightarrow b = 1/2

    After substitution of b into (1):

     0.5a = \sqrt{a}

     0.25a^2 - a = 0

     a(0.25a - 1) = 0 \Longrightarrow a = 0, 4
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  2. #2
    Banned
    Joined
    Aug 2008
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    Quote Originally Posted by scherz0 View Post
    Hello everyone,

    Could someone pleae check my solution for the following problem?

    Thank you!

    ---

    1. Let  a and  b be real numbers with  a > 1 and  b > 0 . If  ab = a^b and  \frac{a}{b} = a^{3b} , determine the value of  a  .



    ---

    When the first equation is multiplied by a:

     b = a^{b - 1} --(3)

    After substitution of (3) in the second equation:

     \frac{a}{a^{b - 1}} = a^{3b}

     a^{2 - b} = a^{3b}

    Therefore:

     2 - b = 3b \Rightarrow b = 1/2

    After substitution of b into (1):

     0.5a = \sqrt{a}

     0.25a^2 - a = 0

     a(0.25a - 1) = 0 \Longrightarrow a = 0, 4
    yes, you are right. Also,

    since, a > 1

    so, a = 4 and b = 1/2 is the solution
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  3. #3
    Junior Member
    Joined
    Jul 2008
    Posts
    74
    Thank you for your reply, shyam and for noting that  a > 1 as I had forgotten!
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