3. In the decimal representation of the number $2^{2017}$, there are $m$ digits, and the number $5^{2017}$ has $n$ digits. Determine the sum $m+n$.
Solution. If the decimal representation of the number $x$ has $m$ digits, then the inequalities are correct

$$
m-1<\log x<m
$$

Thus,

$$
m-1<\log 2^{2017}<m \text { and } n-1<\log 5^{2017}<n .
$$

By adding these inequalities, we get

$$
m+n-2<\log 2^{2017}+\log 5^{2017}<m+n, \text { i.e. } m+n-2<2017<m+n
$$

from which we conclude that $2017=m+n-1$, i.e., $m+n=2018$.