Math

Which of the following functions grows the fastest as x grows without bound?

f(x) = x10
g(x) = ln(x10)
h(x) = 10x
They all grow at the same rate.

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  1. Do you mean?
    f(x) = x^10
    g(x) = ln(x^10)
    h(x) = 10x
    They all grow at the same rate.
    if so
    if x = 1{1^10=1,ln 1^10=0 , 10*1 = 10}

    if x = 10{10^10 =10billion, ln10^10 = 10ln 10 = 23 , 10*10 = 100

    x^10 won hands down

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    posted by Damon
  2. I will assume f(x) = x^10
    also note that g(x) = ln(x^10) = 10lnx

    Not sure if you know Calculus, so I will take an intuitive approach
    Why not look at their graphs?
    http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E10,+y+%3D+10ln(x)+,+y+%3D+10x

    notice that h(x) is just a straight line and rises at a constant rate.
    the ln function grows rather slowly, and is only defined for x > 0
    the f(x) really takes off

    So as x gets bigger , f(x) is the clear winner,
    we could take a look at the race between the other two for some x
    say, x = 5
    f(5) = 5^10 = 50,000,000,000 , rather big
    g(x) = 10ln5 = a mere 16.09..
    h(x) = 50 , coming in 2nd place

    x = 100
    f(100) = 100^10 = 10^20 , really big
    g(100) = 10ln(100) = 46.05..
    h(x) = 500 , still running 2nd

    x = 5000
    f(5000) = not even worth doing, we know that's big
    g(5000) = 85
    h(x) = 50,000

    looks like g(x) < h(x) < f(x)

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    posted by Reiny
  3. yes i did mean that sorry

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    posted by Youngin
  4. Thank you

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    posted by Youngin

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