How many numbers has 2 to the 100th?

How many numbers has 2^{100}?

We put k-as the numbers of the number 2^{100}, Then we have:2^{100}=a\cdot 10^{k-1} where  1<a<10.

By taking the decimal logs in both sides we have:

100log2=loga+(k-1) \Rightarrow k=1+100log2-loga (0<log a<1). k  is the whole number 1+100log2. As we know that the  log2=0.30, we find  k=31 numbers.