AMC 12 2024b Problem 6

The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)

$\textbf{(A) }18 \textbf{(B) }20 \textbf{(C) }22 \textbf{(D) }24 \textbf{(E) }26$