For example, seek for a coin counterfeiter in all coins

Essentially, this is binary search

If you need assume that n is a power of 2, so you have to use binary search like seeking for a coin counterfeiter 

T(n)=T(n/2)+c

first, n

then, n/2

third, n/4

...

last, n/2^k

and, n/2^k=1

so k=

so O(logn)     and it is recurrence

And Master theorem is that T(n)=T(n/2)+c

and n^ = n^=n^0=1    O(1)

and f(n)=c  O(1)

so f(n)= n^, and O(n) is that logn*1     so O(logn)