Question

A line $x=k$ intersects the graph of $y={\log }_{5}x$ and the graph of $y={\log }_5(x+4)$. The distance between the points of intersection is 0.5 . Given $k=a+\sqrt{b}$, where $a$ and $b$ are integers, the value of $(a+b)$ is

• A. 5
• B. 6
• C. 7
• D. 10

Answer 1

Answer: (B)

Obvious $y={\log }_5(x+4)$ is above $y={\log }_{5}x$.
Hence for $x=k,{\log }_5(k+4)-{\log }_{5}k=\frac{1}{2}$

${\log }_5\left(\frac{k+4}{k}\right)=\frac{1}{2}$
$\frac{k+4}{k}=\sqrt{5}$
$1+\frac{4}{k}=\sqrt{5}$
$k=\frac{4}{\sqrt{5}-1}=\sqrt{5}+1\Rightarrow a=1;b=5$
$\therefore a+b=6$