Help with arithmetic in calc class please

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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 thankyou
 
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i got it, nevermind
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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