Solving $5^n > 4,000,000$ without a calculator
\begin{eqnarray} & 5^n &>& 4.000.000\\ \Leftrightarrow & 5^n &>& 5^6 \cdot 2^8 \\ \Leftrightarrow & 5^{n-6} &>& 256.\\ \end{eqnarray} Then, $n=10$.
Divide 4000000 by 5, without a calculator, getting 800000. Divide again; 160000. Again; 32000. Then 6400, then 1280, then 256, then 51 (rounding), then 10, then 2. So $2\times5^9$ is about 4000000, so $5^{10}$ exceeds 4000000.