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Math Help - how much paper would I need to write out this number?? (applied use of logarithms)

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    how much paper would I need to write out this number?? (applied use of logarithms)

    Got this problem to do....... don't know where to start. Need help!

    If 9^(99) is multiplied out and typed on a strip of paper, 3 digits per centimeter, about how many kilometers long would the paper be?

    A hint says use the common log to determine the number of digits the number has, but I don't understand what that means
    Last edited by zachd77; December 27th 2012 at 07:33 PM.
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    Re: how much paper would I need to write out this number?? (applied use of logarithms

    The common log is base 10.
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    Re: how much paper would I need to write out this number?? (applied use of logarithms

    Yes I know that, but I forgot to mention one thing, I revised my post so I'm a bit clearer.
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    Re: how much paper would I need to write out this number?? (applied use of logarithms

    If there are x digits in 9^{9^9} , then solve for x in 10^{x-1}=9^{9^9}.

    Remember to round your value of x down, because you cannot have part of a digit.
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    Re: how much paper would I need to write out this number?? (applied use of logarithms

    So I'd solve for x and then use a conversion factor to get the length of the paper? Also, where does the 10^(x-1) come from?
    Last edited by zachd77; December 27th 2012 at 08:08 PM.
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    Re: how much paper would I need to write out this number?? (applied use of logarithms

    Quote Originally Posted by zachd77 View Post
    So I'd solve for x and then use a conversion factor to get the length of the paper? Also, where does the 10^(x-1) come from?
    10^{x-1} = 9^{9^9} \implies x-1 = 9^9 \log_{10}(9) \approx 369693099, (rounding down to the nearest whole number).

    Add 1 to both sides, and we get

    x=369693100.

    Three digits per centimeter means you will need

    \frac{369693100}{3} = 123231033.\overline{33} \text{ centimeters.}

    Converting from cm to km, the answer is approximately 1232.31 km.
    Thanks from zachd77
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    Re: how much paper would I need to write out this number?? (applied use of logarithms

    Wow.. thanks again!!
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