Logarithmic scale — decibels, Richter, pH — Hinglish notes

3.2.13 · Maths › Exponentials & Logarithms

WHY logarithmic scale use karein?

Ek line mein HOW kaam karta hai: ek reference value $I_{0}$ chuno, phir report karo $L = k lo g_{10} (\frac{I}{I _{0}})$ jahan $k$ "ek step ka size" set karta hai. Alag-alag fields $k$ aur $I_{0}$ alag choose karte hain — decibels, Richter, aur pH mein sirf yehi difference hai.

Teen famous scales

Figure — Logarithmic scale — decibels, Richter, pH

"Step" rules scratch se derive karna

$Δ L = k lo g_{10} r$ apply karna:

Scale	$k$	Quantity ko $r$ se multiply karo	Scale mein change
Decibel	10	×10 (intensity)	$+ 10$ dB
Decibel	10	×2 (intensity)	$+ 10 lo g_{10} 2 \approx + 3$ dB
Richter	1	×10 (amplitude)	$+ 1$
pH	$- 1$	×10 ( $[H^{+}]$ )	$- 1$

Invert karna: scale value se quantity wapas laana

Worked examples

Common mistakes

Recall Feynman: ek 12-saal ke bachche ko explain karo

Socho ek volume knob jahan har click loudness add nahi karta balki multiply karta hai — ek click, do guna loud; ek aur click, phir do guna loud. Numbers bahut tezi se bade hote hain (2, 4, 8, 16, 32…). Ek log scale bas clicks count karta hai giant numbers ki jagah. "60 decibels" matlab "itne clicks upar un quietest sound se jo tum sun sakte ho." Earthquakes ke liye bhi same (har number = 10× shaking) aur acids ke liye bhi (har pH step = 10× acid strength). Yeh ek trillion-fold range ko ek chhote ruler par fit karne ka tarika hai.

Active recall

General log-scale form

L = k lo g_{10} (I / I_{0})

— reference

I_{0}

, step-size constant

k

Decibel formula

L_{dB} = 10 lo g_{10} (I / I_{0})

, with

I_{0} = 1 0^{- 12}

W/m².

Richter formula

M = lo g_{10} (A / A_{0})

; each +1 means ×10 amplitude.

pH formula

pH = - lo g_{10} [H^{+}]

(concentration in mol/L).

Change in level when quantity ×r

Δ L = k lo g_{10} r

(independent of starting value).

+3 dB means what?

Intensity roughly doubled, since

10 lo g_{10} 2 \approx 3.01

Amplitude ratio, magnitude 7 vs 5

1 0^{7 - 5} = 100

times larger.

Invert dB to intensity

I = I_{0} 1 0^{L /10}

Invert pH to concentration

[H^{+}] = 1 0^{- pH}

Why a minus sign in pH?

[H^{+}]

chhota hota hai (e.g.

1 0^{- 3}

) isliye uska log negative hai; minus pH ko positive banata hai.

Can you add decibels directly?

Nahi — physical intensities add karo, phir

10 lo g_{10}

lo.

pH 3 vs pH 6 acidity ratio

1 0^{6 - 3} = 1000

times more acidic at pH 3.

Connections

Laws of Logarithms — product/quotient rules jo yahan har derivation mein kaam aate hain.
Exponential Functions — inverse operation jo ek scale ko un-log karne ke liye use hoti hai.
Change of Base Formula — kyun in scales ke liye base-10 choose kiya jaata hai.
Weber–Fechner Law — decibels ke peeche ka perception principle.
Orders of Magnitude — log scales bas "powers of ten count karna" hain.