Find the number of functions $f(x),$ taking positive real numbers to real numbers, such that
\[f(xy) = f(x) f(y) - 2\]for all positive real numbers $x$ and $y.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Find the number of functions $f(x),$ taking positive real numbers to real numbers, such that
\[f(xy) = f(x) f(y) - 2\]for all positive real numbers $x$ and $y.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
2
Find the number of functions $f(x),$ taking positive real numbers to real numbers, such that
\[f(xy) = f(x) f(y) - 2\]for all positive real numbers $x$ and $y.$
train