Tim and Tom are fictional characters (cats, I believe) who like to play the LoL number game.
In this game, Tim chooses two distinct integers A and B, both ≥ 2, and shows them to Tom. Tom's goal is to find a succession of translolmations that can be applied to A in order to obtain B. There are only two types of translolmations, denoted by the letters $\bf L$ and $\bf o$, in the order as they are applied, from left to right: $\bf L$ replaces the current integer with its square, and $\bf o$ replaces the current integer with the number of digits in its binary representation.
For example, if Tim chooses A = 7 and B = 11, a valid solution for Tom would be $\bf LoLLo$:
$7\buildrel\bf L\over\longrightarrow 49\buildrel\bf o\over\longrightarrow 6\buildrel\bf L\over\longrightarrow 36\buildrel\bf L\over\longrightarrow 1296\buildrel\bf o\over\longrightarrow 11$
For A = 10 and B = 9, valid solutions are $\bf ooL$ and $\bf oLLo$:
$10\buildrel\bf o\over\longrightarrow 4\buildrel\bf o\over\longrightarrow 3\buildrel\bf L\over\longrightarrow 9$
$10\buildrel\bf o\over\longrightarrow 4\buildrel\bf L\over\longrightarrow 16\buildrel\bf L\over\longrightarrow 256\buildrel\bf o\over\longrightarrow 9$
Tom is confident he will always be able to win the game, given a sufficient number of translolmations, no matter what integers Tim chooses. Can you explain how?