Step 1: Behaviour at the ends:
At $x = 2$: $\dfrac{\log2}{2} \approx 0.347$. At $x = e$: $\dfrac1e\approx0.368$. As $x\to\infty$, $\dfrac{\log x}{x}\to 0^+$.
Step 2: Shape:
The value rises slightly from $x=2$ to $x=e$, then falls toward $0$ without ever touching it.
Step 3: Result:
Since the lower bound $0$ is never reached and the open end at $x=2$ is excluded, there is no minimum, option (D).
Final Answer:
There is no minimum value on the interval.
\[ \boxed{\text{(D) }\text{Does not exist}} \]