We are given $n$ identical looking coins, and a scale that allows comparing the weights of two sets of coins $($i$.$e$.,$ we can put any number of coins on each side of the scales$).$ We know that there is exactly one fake coin $-$ it is lighter than other coins. Prove that we can discover the fake coin using $|$~$lo{g}_3(n)$~$|$ comparisons.

$($Example: When $n=3,$ we can discover the fake coin with one comparison. Take any pair of coins and compare their weight: if their weight is identical, then the third coin is the fake one; and otherwise, one coin is lighter than the other, so it must be fake.$)$