Digit Function

We note that $f(x) = \left \lfloor \log_{10} x \right \rfloor + 1$ , where $\lfloor \cdot \rfloor$ denotes the floor function . Then, we have:

$\begin{aligned} f(2^{2019}) + f(5^{2019}) & = \left \lfloor \log_{10} 2^{2019} \right \rfloor + 1 + \left \lfloor \log_{10} 5^{2019} \right \rfloor + 1 \\ & = \left \lfloor 2019 \log_{10} 2 \right \rfloor + \left \lfloor 2019 \log_{10} 5 \right \rfloor + 2 \\ & = \left \lfloor 2019 \times 0.3010 \right \rfloor + \left \lfloor 2019 \times 0.6990 \right \rfloor + 2 \\ & = \left \lfloor 607.719 \right \rfloor + \left \lfloor 1411.281 \right \rfloor + 2 \\ & = 607 + 1411 + 2 = \boxed{2020} \end{aligned}$