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Math Help - Help with arithmetic pleasee

  1. #1
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    Help with arithmetic pleasee

    Given the approximate values: ln(2)=0.7 and ln(10)=2.3
    Use these approximations and an indirect argument using only arithmetic to explain how many decimal digits to expect in the number 2^40 and how do you know your correct?

    Well i know that we'll know if we are correct based on the decimal digits the calculator provides but how do I go about solving for the number of digits??I would greatly appreciate any help
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  2. #2
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    Quote Originally Posted by coe236 View Post
    Given the approximate values: ln(2)=0.7 and ln(10)=2.3
    Use these approximations and an indirect argument using only arithmetic to explain how many decimal digits to expect in the number 2^40 and how do you know your correct?

    Well i know that we'll know if we are correct based on the decimal digits the calculator provides but how do I go about solving for the number of digits??I would greatly appreciate any help
    Hello,

    the number 10^x has x zeros and x+1 digits.

    If \frac{2^{40}}{10^x}=1 then the power to the base 2 is equal to the power with base 10 and must have the same number of digits:

    \frac{2^{40}}{10^x}=1~\implies~40 \cdot \ln(2) - x \cdot \ln(10) = 0 . Use the approximations:

    40 \cdot \ln(2) - x \cdot \ln(10) = 0 ~\implies~28 - x \cdot 2.3 = 0~\implies~x \approx 12.2

    Thus the number 2^{40} has 13 digits.
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