Numerically
2^k=10^n =A that is n=k*log(2) by log laws.
Pick n=1, A=10
2^3=8, 2^4=16
1/4<log(2)<1/3
halving the interval:
7/24<log(2)<1/3
Pick n=2, A=100
2^6=64, 2^7=128
2/7<log(2)<1/3
halving the interval
7/24<log(2)<13/42
Pick n=3, A=1000
2^9=512, 2^10=1024
3/10<log(2)<1/3
From here we...