2014-II-32

Ans: C
I is false. $b>c$.

II is true. According to the given figure, $b$ and $c$ are both greater than $1$. Hence, $bc>1$.

III is true. Note that $B$ and $C$ are on $L$, then the $y$ coordinates of $B$ and $C$ are the same. Let $B=(x_1,y)$ and $C=(x_2,y)$. Hence, we have

$\begin{array}{rcl}
y & = & b^{x_1} \\
x_1 & = & \log_b y
\end{array}$

Also,

$\begin{array}{rcl}
y & = & c^{x_2} \\
x_2 & = & \log_c y
\end{array}$

Hence, we have

$\begin{array}{rcl}
\dfrac{AB}{AC} & = & \dfrac{x_1}{x_2} \\
\dfrac{AB}{AC} & = & \dfrac{\log_b y}{\log_c y} \\
\dfrac{AB}{AC} & = & \dfrac{\frac{\log y}{\log b}}{\frac{\log y}{\log c}} \\
\dfrac{AB}{AC} & = & \dfrac{\log c}{\log b} \\
\dfrac{AB}{AC} & = & \log_b c
\end{array}$