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Thread: Alg II / Pre-Calc w/ Logs: Given log_x(y)+log_y(x)=3, find log_x(y^2)+log_y(x^2)

  1. #1

    Alg II / Pre-Calc w/ Logs: Given log_x(y)+log_y(x)=3, find log_x(y^2)+log_y(x^2)

    Hi there.

    The question I have to complete is this : "Let x and y be two positive real numbers such that logxy+logyx = 3. Find the value of logxy2+logyx2." Now, I have simplified the equation down to 2logx(y)+2logy(x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

    Thanks!

  2. #2
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    Quote Originally Posted by kikithecat View Post
    Hi there.

    The question I have to complete is this : "Let x and y be two positive real numbers such that logxy+logyx = 3. Find the value of logxy2+logyx2." Now, I have simplified the equation down to 2logx(y)+2logy(x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

    Thanks!
    2logx(y)+2logy(x) ......... factor out 2

    = 2 * [logx(y)+logy(x)] ...... continue....
    “... mathematics is only the art of saying the same thing in different words” - B. Russell

  3. #3
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    Quote Originally Posted by kikithecat View Post
    Hi there.

    The question I have to complete is this : "Let x and y be two positive real numbers such that logxy+logyx = 3. Find the value of logxy2+logyx2." Now, I have simplified the equation down to 2logx(y)+2logy(x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

    Thanks!
    Don't square; just factor out the 2!

  4. #4
    Quote Originally Posted by Subhotosh Khan View Post
    2logx(y)+2logy(x) ......... factor out 2

    = 2 * [logx(y)+logy(x)] ...... continue....
    How would the 3 in the original equation be integrated into this? Would it be squared, would it be left as 3...?

  5. #5
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    Quote Originally Posted by kikithecat View Post
    Hi there.

    The question I have to complete is this : "Let x and y be two positive real numbers such that logxy+logyx = 3. Find the value of logxy2+logyx2." Now, I have simplified the equation down to 2logx(y)+2logy(x), but I don't know how to find the value after that. Would the 3 from the original equation have to be squared as well? Any guidance from there would be helpful.

    Thanks!
    Try it this way.

    [tex]\text {GIVEN: } x,\ y \in \mathbb R^+ \text { and } log_x(y) + log_x(y) = 3.[/tex]

    [tex]\text {Let } u = log_x(y) \text { and } v = log_y(x) \implies[/tex]

    [tex]u + v = 3 \text { and } log_x(y^2) = 2u \text { and } log_y(x^2) = 2v \implies [/tex]

    [tex]log_x(y^2) + log_y(x^2) = 2u + 2v = 2(u + v) = WHAT?[/tex]

  6. #6
    Quote Originally Posted by JeffM View Post
    Try it this way.

    [tex]\text {GIVEN: } x,\ y \in \mathbb R^+ \text { and } log_x(y) + log_x(y) = 3.[/tex]

    [tex]\text {Let } u = log_x(y) \text { and } v = log_y(x) \implies[/tex]

    [tex]u + v = 3 \text { and } log_x(y^2) = 2u \text { and } log_y(x^2) = 2v \implies [/tex]

    [tex]log_x(y^2) + log_y(x^2) = 2u + 2v = 2(u + v) = WHAT?[/tex]
    Oh, I see now! Using a simpler version of the problem was helpful. I was over-analyzing it, believing that I needed to find the values of x and y when it was really as simple as that. Thank you!

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