1. ## 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

2. Originally Posted by coe236
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.