Escalating twos

$0.30 < \log_{10} 2 < 0.31 \\ 30 < 100\log_{10} 2 < 31 \\ 30 < \log_{10} 2^{100} < 31 \\ 10^{30} < 2^{100} < 10^{31}$

Hence $2^{100}$ has the same amount of digits $10^{30}$ which is $\boxed{31}$

Using Logarithms

in a logarithm number ,it comprises of two things. The characteristic and the mantissa. example. in 2.1452 2 is the characteristic and 1452 is the mantissa

The characteristic specifies the length of the integer, and the mantissa, it's logarithmic value. Hence if we see 100.something, the integer is (100+1) digits long. Therefore

in 2^100 , after taking log, it becomes 100log(2)=100x(0.3010)=30.10 therefore the number is 31.1 ~=31 digits long [taking the approximation].

FROM 2^10....FOR 20 30 40.......100 THE COMMON DIFFERENCE BETWEEN THE NUMBER OF DIGITS IS 3.