Two horizontal rulers show the same number v, starting at 16. The top ruler is a plain linear scale from 0 to 1024; the bottom is a log ruler where the powers of two 1, 2, 4 up to 1024 are spaced equally. Pressing times-2 doubles v: on the plain ruler each jump is twice as wide as the last, while on the log ruler every jump is exactly one tick. This shows that log of v simply counts how many multiplies reach v, which turns multiplication into addition.
The log counts multiplies
One number, two rulers. Press × 2 a few times: on the plain ruler each jump is twice as wide as the last — on the log ruler every jump is exactly one tick.
value v16
doublings log₂(v)4
in ×10 steps log₁₀(v)1.20
multiply the value → add one tick: log₂(v × 2) = log₂ v + 1
The log isn’t a formula to memorize — it answers one question: “how many multiplies to reach v?” That single move turns multiplying into adding: multiply the value, add one tick — log(a×b) = log a + log b. The base (2, 10, e) only changes the size of one tick, like inches vs centimeters on the same ruler.