Change of base formula — proof — Hinglish notes

3.2.9 · Maths › Exponentials & Logarithms

Formula kya kehta hai

HAR piece kya hai:

$a$ = purana base (jise tum hatana chahte ho).
$b$ = naya base (jo tumhara calculator samajhta hai, jaise $10$ ya $e$ ).
$x$ = argument (andar wala number).

Scratch se derivation (kabhi memorise mat karo — khud banao)

HUM ise step by step kaise derive karte hain:

Step 1 — Jis cheez ko chahte hain usse naam do. Maano $y = lo g_{a} x .$ Yeh step kyun? Ek mystery ko manipulate nahi kar sakte; use naam $y$ dene se hum uspe algebra kar sakte hain.

Step 2 — Exponential form mein likhte hain. Upar wali definition se, $a^{y} = x .$ Yeh step kyun? Exponentials par kisi bhi base ke logs se attack karna aasaan hota hai, kyunki logs powers ko products mein badal dete hain.

Step 3 — DONO sides par $lo g_{b}$ lete hain. $lo g_{b} (a^{y}) = lo g_{b} x .$ Yeh step kyun? Dono sides par same function apply karne se equality bani rehti hai. Hum base $b$ isliye choose karte hain kyunki yahi woh base hai jo hum use kar sakte hain (calculator base).

Step 4 — Power law $lo g_{b} (a^{y}) = y lo g_{b} a$ use karte hain. $y lo g_{b} a = lo g_{b} x .$ Yeh step kyun? Log lene ka yahi poora point hai: yeh exponent $y$ ko neeche kheench ke ek coefficient bana deta hai, taaki hum use isolate kar sakein.

Step 5 — $y$ ke liye solve karo ( $lo g_{b} a$ se divide karo, jo $\neq = 0$ hai kyunki $a \neq = 1$ ): $y = \frac{l o g _{b} x}{l o g _{b} a} .$

Step 6 — Yaad karo $y$ kya tha. Kyunki $y = lo g_{a} x$ , $lo g_{a} x = \frac{lo g _{b} x}{lo g _{b} a} ■$

Worked examples

Common mistakes (Steel-manned)

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

Ek log ek sawaal hai: "Kitni baar is base ko multiply karna padega taaki yeh number mile?" $lo g_{2} 8$ poochta hai "kitne 2s multiply karke 8 banta hai?" — jawab 3 hai. Ab imagine karo tumhare paas sirf ek base-10 ruler hai powers measure karne ke liye. Phir bhi tum "2-wala sawaal" maap sakte ho — bas maap lo $8$ kitna bada hai base 10 mein, maap lo $2$ kitna bada hai base 10 mein, aur divide karo. Scaling cancel ho jaati hai aur tumhe sahi jawab milta hai, 3. Same height, alag ruler.

Active recall

Definition

lo g_{a} x = y

ka exponential form mein kya matlab hai?

a^{y} = x

Change of base formula state karo.

lo g_{a} x = \frac{lo g _{b} x}{lo g _{b} a}

Change of base mein numerator mein kaunsi quantity jaati hai?

Argument

x

ka log (new base mein).

Denominator mein kaunsi quantity jaati hai?

Old base

a

ka log (new base mein).

lo g_{b} a

se divide karna allowed kyun hai?

Kyunki

a \neq = 1

, isliye

lo g_{b} a \neq = 0

Proof ka pehla step?

y = lo g_{a} x

maano, phir

a^{y} = x

likh do.

Proof mein exponent neeche laane wala kaunsa log law hai?

Power law

lo g_{b} (a^{y}) = y lo g_{b} a

Reciprocal identity

lo g_{a} b = 1/ lo g_{b} a

prove karo.

x = b

rakho:

lo g_{a} b = \frac{l o g _{b} b}{l o g _{b} a} = \frac{1}{l o g _{b} a}

Base 10 se

lo g_{2} 8

compute karo.

\frac{l o g 8}{l o g 2} = 3

Kya new base sirf 10 ya

e

hi hona chahiye?

Nahi — koi bhi valid base

b > 0, b \neq = 1

kaam karta hai; 10/

e

sirf convenient hain.

lo g_{a} b \cdot lo g_{b} a

simplify karo.

= 1

(yeh dono reciprocals hain).

Connections

Definition of a logarithm — woh ek fact jis par poora proof tika hai.
Laws of logarithms — power law derivation ka engine hai.
Exponential functions — Step 2 mein log→exponential form mein convert hota hai.
Natural logarithm ln — common "new base" $b = e$ .
Solving exponential equations — change of base unknown exponents isolate karne deta hai.

Formula kya kehta hai

Scratch se derivation (kabhi memorise mat karo — khud banao)

Worked examples

Common mistakes (Steel-manned)

Active recall

Connections

Concept Map