Logarithmic scale — decibels, Richter, pH: Visual walkthrough

Step 1 — The problem: numbers that explode

Figure — Logarithmic scale — decibels, Richter, pH

Here $I$ = the actual intensity, and the tick marks are spaced by equal additions ( $+ 0.2$ , $+ 0.4$ , …). That is a linear scale, and it clearly fails us.

Step 2 — The clue: our senses multiply, they don't add

Symbols so far: $I_{0}$ = a chosen reference starting intensity; $r$ = the fixed multiplier per click.

Step 3 — Inventing the log: "how many clicks?"

In $L = lo g_{10} (I / I_{0})$ : the piece $\frac{I}{I _{0}}$ (blue) says "how many times bigger than the reference"; the $lo g_{10}$ (magenta) says "how many powers of ten is that".

Step 4 — The general scale, term by term

Step 5 — The central result: why a step depends only on the ratio

This is the heart of the whole topic. We prove that changing $I$ by a factor $r$ always shifts $L$ by the same amount, no matter where you started.

Step 6 — Reading off the three famous scales

Intuition WHAT / WHY / PICTURE

WHAT. Plug the field's choice of $k$ into $Δ L = k lo g_{10} r$ :

Scale	$k$	×10 gives	×2 gives
Decibel	$10$	$+ 10$ dB	$10 lo g_{10} 2 \approx + 3.01$ dB
Richter	$1$	$+ 1$	$+ 0.301$
pH	$- 1$	$- 1$	$- 0.301$

WHY the pH minus. Hydrogen-ion concentration $[H^{+}]$ is a tiny number (like $1 0^{- 3}$ ), so its log is negative. Choosing $k = - 1$ flips that into a friendly positive number, and — crucially — makes stronger acid read lower.

PICTURE. Three rulers stacked, sharing one "×10" step, showing how $k$ rescales it: $+ 10$ , $+ 1$ , and (flipped) $- 1$ .

Step 7 — The degenerate & edge cases (never get surprised)

Intuition WHAT / WHY / PICTURE

Every case the formula could face:

$I = I_{0}$ (at the reference). Then $I / I_{0} = 1$ and $lo g_{10} (1) = 0$ , so $L = 0$ . The ruler's origin. For dB this is the threshold of hearing (0 dB — not silence!).
$I < I_{0}$ (below reference). Then $I / I_{0} < 1$ , its log is negative, so $L$ is negative. Sounds quieter than the threshold have negative decibels. Perfectly valid, just left of zero.
$I \to 0$ (silence). $lo g_{10}$ of something approaching $0$ dives to $- \infty$ . True silence is $- \infty$ dB — this is why "0 dB" is not silence; the scale has no bottom.
$r = 1$ (no change). $Δ L = k lo g_{10} 1 = 0$ . Nothing moved — reassuring.
$I < 0$ ? Intensity, amplitude and concentration are never negative, so $I / I_{0} > 0$ always — the log is always defined. Good: the physics protects the maths.

PICTURE. The full ruler extending to $- \infty$ on the left, with the origin marked at $I_{0}$ and the "cannot happen" negative region greyed out.

Step 8 — Inverting: from ruler value back to the quantity

The figure shows the two staircases: log going up (big number → small label) and exponential coming back down (label → big number), meeting at the reference.

The one-picture summary

Everything on one canvas: the exploding linear numbers up top, the neat log ruler below, the "×r ⇒ + $k lo g_{10} r$ " arrow that is the same length everywhere, and the three scales branching off by their choice of $k$ .

Recall Feynman retelling — the whole walkthrough in plain words

Real quantities like loudness, shaking, and acidity spread over a trillion-fold range, so a normal ruler is useless — everything piles up at one end. But here is the secret: our senses and nature don't add, they multiply — turn a knob one click and the sound doubles, not "goes up by 5". So instead of writing the giant numbers, we just count the clicks. Counting clicks is the logarithm. Because each click is a fixed multiply, every "×10 jump" takes the exact same number of clicks no matter where you start — that's why the ruler comes out evenly spaced and why "×r always shifts the value by $k lo g_{10} r$ ". Finally we pick a step-size $k$ for each field ( $10$ for decibels, $1$ for Richter, $- 1$ for pH to flip tiny acid numbers upright), and to go back to real numbers we just replay the clicks with a power of ten. That's the entire scale, built from a volume knob.

Active recall

Connections

Laws of Logarithms — the product rule that made Step 5 possible.
Exponential Functions — the inverse used in Step 8 to un-log a value.
Change of Base Formula — why base-10 is a free convenience choice (Step 3).
Weber–Fechner Law — the "senses multiply" clue of Step 2.
Orders of Magnitude — each log unit is one power of ten.