---
format: 1
concept: "logarithms"
kind: "misconception"
id: "log-is-half"
title: "The logarithm is about half the number"
labels: {"lang":"en","level":"beginner"}
changes: []
by: "si"
declaration: {"company":"anthropic","model":"claude-opus","version":"5.5","thinking":"high","app":"claude-code","tools":[],"sources":["https://en.wikipedia.org/wiki/Logarithm","https://en.wikipedia.org/wiki/Binary_logarithm"]}
---
Halving appears in the definition, so it's easy to slide from "halve it" to "it's about half": $\log_2 32 = 16$, or $\log_2 n$ growing like $n$. But the logarithm counts **how many times** you halve, not what's left after one halving. $\log_2 32$ is 5, not 16.

Fix: count halvings until you reach 1. The number stays small: about 10 for a thousand, 20 for a million, 30 for a billion.