Core Idea - Binary Search for LCA

Once $a$ and $b$ are at the same depth, jump both upward simultaneously in powers of $2$. For each $j$ from $\log n$ down to $0$, check if up[a][j] != up[b][j].

If so, jump both: a = up[a][j], b = up[b][j]. After all jumps, $a$ and $b$ are directly below the LCA. The LCA is up[a][0] (or up[b][0], same thing). You try large jumps first and only make a jump if it does not overshoot the LCA.