1/logx xy + 1/logy xy A) 0 B) 1 C) -1 D) 2

Question

\(\cfrac{1}{log_{x}xy}\) + \(\cfrac{1}{log_{y}xy}\)

1/log_x xy + 1/log_y xy 

A) 0 

B) 1 

C) -1 

D) 2

Answer 1

Correct option is (B) 1

\(\frac{1}{log_x\,xy} + \frac{1}{log_y\,xy} \) \(=\frac{1}{log_x\,x+log_x\,y} + \frac{1}{log_y\,x+log_y\,y} \)   \((\because\) log AB = log A + log B)

\(=\frac{1}{1+log_x\,y} + \frac{1}{1+log_y\,x} \)   \((\because log_aa=1)\)

\(=\frac{1}{1+log_x\,y} + \frac{1}{1+\frac1{log_x\,y}} \)    \((\because log_a\,b=\frac1{log_b\,a})\)

\(=\frac{1}{1+log_x\,y} + \frac{log_x\,y}{1+log_x\,y} \)

\(=\frac{1+log_x\,y}{1+log_x\,y}=1\)

Answer 2

Correct option is B) 1