Logarithmic scale — decibels, Richter, pH

WHY use a logarithmic scale at all?

HOW it works in one line: pick a reference value $I_0$ , then report $L = k\,\log_{10}\!\left(\frac{I}{I_0}\right)$ where $k$ sets the "size of one step." Different fields choose $k$ and $I_0$ differently — that's the only difference between decibels, Richter, and pH.

The three famous scales

Figure — Logarithmic scale — decibels, Richter, pH

Deriving the "step" rules from scratch

Formula Derivation: what does "+1 unit" mean?

Start from the general form $L = k\log_{10}(I/I_0)$ . Suppose the underlying quantity multiplies by a factor $r$ : $I \to rI$ . Then the new level is $L' = k\log_{10}\!\left(\frac{rI}{I_0}\right).$ Why this step? Use $\log(rI/I_0)=\log r + \log(I/I_0)$ (product rule) to separate the multiplier from everything else: $L' = k\log_{10} r + \underbrace{k\log_{10}\!\left(\frac{I}{I_0}\right)}_{L}.$ So the change is $\boxed{\;\Delta L = L' - L = k\log_{10} r\;}$ which depends only on the ratio $r$ , not on where you started. This is the whole magic of log scales.

Applying $\Delta L = k\log_{10} r$ :

Scale	$k$	Multiply quantity by $r$	Scale changes by
Decibel	10	×10 (intensity)	$+10$ dB
Decibel	10	×2 (intensity)	$+10\log_{10}2\approx +3$ dB
Richter	1	×10 (amplitude)	$+1$
pH	$-1$	×10 ( $[\text{H}^+]$ )	$-1$

Inverting: going from scale value back to the quantity

Worked examples

Common mistakes

Recall Feynman: explain it to a 12-year-old

Imagine a volume knob where each click doesn't add loudness but multiplies it — click once, twice as loud; click again, twice as loud again. Numbers get huge fast (2, 4, 8, 16, 32…). A log scale just counts the clicks instead of the giant numbers. "60 decibels" means "so many clicks up from the quietest sound you can hear." Same for earthquakes (each number = 10× the shaking) and acids (each pH step = 10× the acid strength). It's a way to fit a trillion-fold range onto a small ruler.

Active recall

General log-scale form

L = k\log_{10}(I/I_0)

— reference

I_0

, step-size constant

k

Decibel formula

L_{\text{dB}} = 10\log_{10}(I/I_0)

, with

I_0=10^{-12}

W/m².

Richter formula

M = \log_{10}(A/A_0)

; each +1 means ×10 amplitude.

pH formula

\text{pH} = -\log_{10}[\text{H}^+]

(concentration in mol/L).

Change in level when quantity ×r

\Delta L = k\log_{10}r

(independent of starting value).

+3 dB means what?

Intensity roughly doubled, since

10\log_{10}2 \approx 3.01

Amplitude ratio, magnitude 7 vs 5

10^{7-5}=100

times larger.

Invert dB to intensity

I = I_0\,10^{L/10}

Invert pH to concentration

[\text{H}^+] = 10^{-\text{pH}}

Why a minus sign in pH?

[\text{H}^+]

is tiny (e.g.

10^{-3}

) so its log is negative; the minus makes pH positive.

Can you add decibels directly?

No — add the physical intensities, then take

10\log_{10}

pH 3 vs pH 6 acidity ratio

10^{6-3}=1000

times more acidic at pH 3.

Connections

Laws of Logarithms — the product/quotient rules that power every derivation here.
Exponential Functions — the inverse operation used to un-log a scale.
Change of Base Formula — why base-10 is chosen for these scales.
Weber–Fechner Law — the perception principle behind decibels.
Orders of Magnitude — log scales are just "counting powers of ten."

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, kuch real-world quantities bahut zyada range cover karti hain — jaise sound intensity ek whisper se lekar jet engine tak trillion guna (10^12) badal jaati hai. Itne bade numbers likhna mushkil hai. Isliye hum logarithmic scale use karte hain: jab bhi asli quantity multiply hoti hai (×10, ×10…), scale ka number sirf add hota hai (+1, +1…). Yahi log ka jaadu hai: $\log(ab)=\log a+\log b$ — multiplication ko addition bana deta hai.

Teenon famous scales bilkul ek hi formula se aate hain: $L=k\log_{10}(I/I_0)$ . Sirf $k$ aur reference $I_0$ change hota hai. Decibel mein $k=10$ (isliye ×2 power = +3 dB, kyunki $10\log 2\approx 3$ ). Richter mein $k=1$ (har +1 magnitude = ×10 shaking, toh M7 vs M5 = $10^2=100$ guna). pH mein $k=-1$ (minus isliye kyunki $[\text{H}^+]$ chhota hota hai jaise $10^{-3}$ , log negative aata, minus usko positive kar deta).

Sabse common galti: decibels ko seedhe add mat karo! 60 dB + 60 dB = 120 dB galat hai. Do same sources ka matlab intensity double, yaani sirf +3 dB → 63 dB. Rule yaad rakho: pehle intensities add karo, phir log lo. Aur Richter mein bhi 8 aur 4 ka ratio 2 nahi hota — woh $10^{8-4}=10000$ guna hota hai. Hamesha subtract-then-power-of-ten karo.

Yeh matter isliye karta hai kyunki humari kaan, zameen, aur chemistry sab "multiplicative" tareeke se behave karte hain. Log scale exactly usi feel ko match karta hai, aur ek trillion-guna range ko ek chhoti si ruler par fit kar deta hai. Exam mein derivation ( $\Delta L=k\log_{10}r$ ) khud likhna aana chahiye — yahi 80/20 wala core hai.

Logarithmic scale — decibels, Richter, pH

WHY use a logarithmic scale at all?

The three famous scales

Deriving the "step" rules from scratch

Inverting: going from scale value back to the quantity

Worked examples

Common mistakes

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Exponentials & Logarithms

Connections