The Inverse of Exponential

Algebra 2 · Axiom Academy

The pH scale hides an exponential, . To run it backward you need a brand-new operation — the logarithm, . Six liquids, a billionfold apart — one tidy scale Acids and bases all come down to one thing: how much hydrogen ion, , is floating in the liquid. Lemon juice holds about a billion times more of it than ammonia. Numbers that lopsided are impossible to line up on an ordinary axis — so chemists built a smarter one. Press play. Six everyday liquids start crushed against a plain concentration axis — four of them pile into a single unreadable smudge. Then the axis rescales so that every tenfold change in becomes one even step. That rescaled axis is pH. One even step of pH always means a tenfold change in acidity — the fingerprint of a logarithm. From pH to concentration: raising ten to a power The pH gives the concentration directly: . Drag the pH and watch it — every single step up in pH divides the acid concentration by ten. On an axis spaced by powers of ten, that steady tenfold fall is a straight line. Each unit of pH is another factor of ten — so the whole scale is a ladder of powers of ten. From concentration to pH: the logarithm undoes it Here is the real question the pH formula answers: given the concentration, what is the pH? Start at 1 mol/L and count how many times you divide by ten to reach . That count is the pH — and counting the tens is exactly what the logarithm does, . Exponentials pack the tens in; logarithms count them back out. Each one undoes the other.

This is the written version of the interactive lesson above. See the full Algebra 2 course.