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Thread: Least Possible Value of k

  1. #1
    Super Member harpazo's Avatar
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    Least Possible Value of k

    If k is an integer and 0.0010101 x 10^(k) is greater than 1,000, what is the LEAST possible value of k?

    Set Up:

    0.0010101 x 10^(k) > 1,000.

    Must I take the log on both sides to find k?
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Least Possible Value of k

    You can eyeball this one as the value of 0 < k will determine how many places the decimal point in the mantissa will move to the right.
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    Re: Least Possible Value of k

    Quote Originally Posted by harpazo View Post
    If k is an integer and 0.0010101 x 10^(k) is greater than 1,000, what is the LEAST possible value of k?

    Set Up:

    0.0010101 x 10^(k) > 1,000.

    Must I take the log on both sides to find k?
    I don't know if you "must" take the logarithm (especially base 10) of each side to find k, but it is geared for it.


    $\displaystyle 0.0010101*10^k > 1,000$

    I would isolate the exponential term first:


    $\displaystyle 10^k \ > \ \dfrac{1,000}{0.0010101}$


    $\displaystyle 10^k \ > 990,000.99 \ \ \ \ \ \ \ \ \ $ (rounded to two decimal places)


    $\displaystyle \log_{10}(10^k) \ > \ \log_{10}(990,000.99)$


    $\displaystyle k \ > \ \log_{10}(990,000.99)$


    What is your conclusion from looking at the result after entering the above in a calculator/computer?
    Last edited by greg1313; Dec 11th 2018 at 02:40 PM.
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    Super Member harpazo's Avatar
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    Re: Least Possible Value of k

    Quote Originally Posted by greg1313 View Post
    I don't know if you "must" take the logarithm (especially base 10) of each side to find k, but it is geared for it.


    $\displaystyle 0.0010101*10^k > 1,000$

    I would isolate the exponential term first:


    $\displaystyle 10^k \ > \ \dfrac{1,000}{0.0010101}$


    $\displaystyle 10^k \ > 990,000.99 \ \ \ \ \ \ \ \ \ $ (rounded to two decimal places)


    $\displaystyle \log_{10}(10^k) \ > \ \log_{10}(990,000.99)$


    $\displaystyle k \ > \ \log_{10}(990,000.99)$


    What is your conclusion from looking at the result after entering the above in a calculator/computer?
    My conclusion is that k = 6.
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    MHF Contributor MarkFL's Avatar
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    Re: Least Possible Value of k

    Quote Originally Posted by harpazo View Post
    My conclusion is that k = 6.
    Yes, that's what I got from "eyeballing" it as I described.
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    Super Member harpazo's Avatar
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    Re: Least Possible Value of k

    Quote Originally Posted by MarkFL View Post
    Yes, that's what I got from "eyeballing" it as I described.
    I needed to work it out to arrive at k = 6.
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