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Interrelationship between power of 2 and integer length

  1. Oct 31, 2013 #1
    Hello, I was wondering if someone knew the equation which describe the exact or statistical relation between the length(# of digits) of a power of 2 based on it's power.

    I plotted 200 of the powers in mathematica and I get a fairly straightforward staircase plot. I'm just wondering what's the rule here.

    4.jpg
     
  2. jcsd
  3. Oct 31, 2013 #2
    The length of an integer ##x## in basis 10 is given by ##\lfloor\log_{10}(x)\rfloor+1##, where the strange brackets denote the largest integer smaller or equal than ##x##. So if ##x=2^n##, then the length is

    [tex]\lfloor \log_{10}(2^n)\rfloor + 1 \sim n\log_{10}(2) + 1[/tex]
     
  4. Oct 31, 2013 #3
    If the number of digits of a number increases linearly, the number increases exponentially; if the exponent of a number increases linearly, the number increases exponentially. Basically you've plotted the implicit function 10^y = 2^x.

    y/x = b is a constant because the equation 2 = 10^b has one root. To estimate b consider 2^10 = 1024 and 10^3 = 1000.
     
    Last edited: Oct 31, 2013
  5. Nov 7, 2013 #4
    n * (memorize 0.3 as log[10,2]) +1, excellent, thanks!
     
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