Common log (log₁₀) and natural log (ln x)

3.2.7 · Maths › Exponentials & Logarithms

WHY kyon yeh do bases matter karti hain?

WHY base 10? Kyunki hum numbers base 10 mein likhte hain. $lo g_{10} (1000) = 3$ literally digits ka order of magnitude count karta hai: $1000 = 1 0^{3}$ . Toh common log measure karta hai ki kitni powers of ten ek number mein hain.

WHY base $e$ ? Kyunki $e^{x}$ woh akela function hai jo apni derivative ke barabar hai: $\frac{d}{d x} e^{x} = e^{x}$ . Yahi cheez $ln x$ ko woh log banati hai jo jab bhi calculus involved ho tab appear karta hai. Hum neeche dekhenge ki $\frac{d}{d x} ln x = \frac{1}{x}$ — ek beautifully clean result jo koi aur base nahi deta.

HOW $e$ kahan se aata hai? (Derivation from scratch)

Hum $e$ ko sirf assume nahi karte. Chaliye ise build karte hain.

Yeh step kyun? Har ek $n$ periods mein tumhara paisa $(1 + \frac{1}{n})$ se multiply hota hai, aur $n$ periods hote hain, toh tum $n$ baar multiply karte ho.

Ab $n \to \infty$ push karo (har ek instant mein compound karo):

$e := lim_{n \to \infty} (1 + \frac{1}{n})^{n} = 2.718281828 \dots$

$n$	$(1 + 1/ n)^{n}$
1	2.000
10	2.594
100	2.705
1000	2.717
$\infty$	2.71828…

Toh $e$ "smoother aur smoother growth" ka limit hai. Isliye $e$ continuous processes govern karta hai.

HOW $\frac{d}{d x} ln x = \frac{1}{x}$ derive karte hain

Derivation. Maano $y = ln x$ , toh $x = e^{y}$ . Dono sides ko $x$ ke w.r.t. differentiate karo: $1 = e^{y} \frac{d y}{d x} ⟹ \frac{d y}{d x} = \frac{1}{e ^{y}} = \frac{1}{x} .$

Yeh step kyun? Humne $e^{y}$ pe chain rule use kiya (jis ki derivative woh khud hai) aur phir $e^{y} = x$ wapas substitute kiya. Koi aur base itna cleanly collapse nahi hota — base 10 ke liye ek factor $ln 10$ aa jaata.

HOW bases ke beech convert karte hain (change-of-base bridge)

Derivation. Maano $y = lo g_{b} x$ , toh $b^{y} = x$ . Dono sides ka $ln$ lo: $ln (b^{y}) = ln x \Rightarrow y ln b = ln x \Rightarrow y = \frac{l n x}{l n b} .$

Yeh step kyun? Humne ek equation jisme hidden exponent $y$ tha, use power law $ln (b^{y}) = y ln b$ lagaake ek linear equation mein badal diya. Yeh master trick hai: logs exponents ko neeche kheench lete hain.

Figure — Common log (log₁₀) and natural log (ln x)

Worked examples

Forecast-then-Verify

Common mistakes (Steel-manned)

Flashcards

lo g_{10} x

kaunsa sawaal answer karta hai?

"10 ko kis power pe raise karo taaki

x

mile?"

e

ki numerical value 4 dp tak kya hai?

2.7183

e

ko ek limit ke roop mein define karo.

e = lim_{n \to \infty} (1 + \frac{1}{n})^{n}

Base

e

calculus mein kyun dominate karta hai?

Kyunki

\frac{d}{d x} e^{x} = e^{x}

, jisse

\frac{d}{d x} ln x = \frac{1}{x}

milta hai.

lo g_{b} x

ka change-of-base formula kya hai?

lo g_{b} x = \frac{ln x}{ln b}

\frac{d}{d x} ln x

kya hai?

\frac{1}{x}

\int \frac{1}{x} d x

kya hai?

ln ∣ x ∣ + C

ln 10

approximately kya hai?

2.3026

Convert karo:

lo g_{10} x

ln

ke terms mein.

lo g_{10} x = \frac{ln x}{ln 10}

e^{3 x} = 20

solve karo.

x = \frac{ln 20}{3} \approx 0.999

lo g x

aur

ln x

ka domain kya hai?

x > 0

(sirf positive reals)

Kya

lo g (a + b) = lo g a + lo g b

Nahi — logs products ko split karte hain, sums ko nahi.

pH ko log ke terms mein kya hota hai?

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

lo g_{10} e

ki value kya hai?

\approx 0.4343 = \frac{1}{ln 10}

Recall Feynman: 12-saal ke bacche ko explain karo

Ek logarithm ek "kitni baar multiply kiya?" ka counter hai. Socho paper ko tens mein fold karo: 10, 100, 1000. Common log sirf zeros count karta hai — $lo g_{10} (1000) = 3$ kyunki 3 tens multiply hue hain. Hamare number system ke liye easy! Natural log ek magic number $e \approx 2.72$ use karta hai jo tab appear hota hai jab cheezein smoothly aur continuously grow karti hain — jaise paisa har ek instant pe interest kama raha ho, ya bacteria nonstop double ho rahe hon. Scientists $e$ pasand karte hain kyunki calculus mein yeh sab neat banata hai: $e^{x}$ ki "slope" bilkul $e^{x}$ hi hoti hai, woh akela function jo khud ki copy karta hai. Toh $ln$ woh log hai jo us magic growth ke saath pair karta hai.

Connections

Laws of Logarithms (product, quotient, power) — woh algebra jo logs ko exponents neeche kheenchne deti hai.
Exponential functions e^x and 10^x — woh inverses jo yeh logs undo karte hain.
Solving exponential equations — jahan $ln$ / $lo g$ choice ke tools hain.
Differentiation of ln x and e^x — kyun base $e$ calculus-natural hai.
Exponential growth and decay models — $ln$ ka real-world ghar.
Scientific notation and orders of magnitude — $lo g_{10}$ ka natural ghar.