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Please show your solution and how you did it. Thank you so much.

1.If x and y are positive numbers such that x^y=y^x and y =2012x, what is x?

2. What powers of two prime numbers will give a product of 1000...000(this sign --> { below the 1000...000 and below this { telling 2012 zeros )

  • Math -

    Please provide the solution, how you did it pleaseeee<3 Thank youuuuuuuuuu

  • Math -

    1.
    x^y = y^x ,and y = 2012x

    x^(2012x) = (2012x)^x
    take ln of both sides, and use log rules

    2012x lnx = x ln(2012x)
    2012x lnx = x (ln2012 + lnx)
    2012 lnx = ln 2012 + lnx
    2012 lnx - lnx = ln2012
    lnx(2012 - 1) = ln2012
    lnx = ln2012 /2011 = .003782637
    x = e^.00378... = 1.003789801
    y = 2012(1.0037...) = 2019.625079

    YOu can check with your calculator, the values work in the original equation x^y = y^x

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