If one uses only the tabular information $10^3=1000$, $10^4=10,000$, $2^{10}=1024$, $2^{11}=2048$, $2^{12}=4096$, $2^{13}=8192$, then the strongest statement one can make for $\log_{10}{2}$ is that it lies between:
$\textbf{(A)}\ \frac{3}{10} \; \text{and} \; \frac{4}{11}\qquad \textbf{(B)}\ \frac{3}{10} \; \text{and} \; \frac{4}{12}\qquad \textbf{(C)}\ \frac{3}{10} \; \text{and} \; \frac{4}{13}\qquad \textbf{(D)}\ \frac{3}{10} \; \text{and} \; \frac{40}{132}\qquad \textbf{(E)}\ \frac{3}{11} \; \text{and} \; \frac{40}{132}$
Since $1024$ is greater than $1000$.
$\log (1024) > 3$
$10 * \log (2) > 3$
and $\log (2) > 3/10$.

Similarly, $8192 < 10000$, so $\log (8192) < 4$
$13 * \log (2) < 4$
and $\log (2) < 4/13$

Therefore  $3/10 < \log 2 < 4/13$ 
so the answer is $\fbox{C}$