6.65 The line $x=k$ intersects the curves $y=\log _{5} x$ and $y=\log _{5}(x+4)$ at two points with a distance of 0.5. If $k=a+\sqrt{b}$, where $a, b$ are integers, then $a+b=$
(A) 6.
(B) 7.
(C) 8.
(D) 9.
(E) 10.
(48th American High School Mathematics Examination, 1997)
[Solution] From the given, we know:
$$
\log _{5}(k+4)-\log _{5} k=0.5,
$$

which means $\frac{k+4}{k}=\sqrt{5}$, leading to $k=\sqrt{5}+1$, thus $a+b=6$. Therefore, the answer is $(A)$.