Question 129
How many digits does the number $3^{100}$ have? Well, we can give an approximate answer to this question, without using a calculator, simply by comparing it with powers of $10$. Note that $3^{2}\lt 10$ allows us to conclude that $(3^{2})^{50}=3^{100}\lt 10^{50}$. Then, $3^{100}$ has at most $50$ digits, since $10^{50}$ is the first number with $51$ digits. The number $3^{100}$ actually has $48$ digits! Now it is your turn. Let $N$ be the number of digits of the number $2^{100}$; determine a positive integer $k$ such that $k \leq N \leq k+5$.