Graphs of logarithmic functions — Hinglish notes

3.2.12 · Maths › Exponentials & Logarithms

LOG FUNCTION kya hoti hai?

$a \neq = 1$ aur $a > 0$ kyun? Kyunki $1^{y} = 1$ har $y$ ke liye (koi unique inverse nahi), aur negative bases bahut se exponents ke liye undefined values dete hain. Exponentials waali hi restriction hai — logs ussi ko inherit karte hain.

Graph kaise banta hai (scratch se derivation)

Hum already jaante hain exponential graph $y = a^{x}$ (jab $a > 1$ ho):

$(0, 1)$ se guzarta hai
horizontal asymptote $y = 0$ hai
hamesha positive, increasing.

$y = lo g_{a} x$ paane ke liye hum $y = x$ mein reflect karte hain (kyunki geometrically "inverse function" ka yehi matlab hai — input aur output swap karo). Kisi point $(p, q)$ ko $y = x$ mein reflect karne se $(q, p)$ milta hai. Toh har feature apna role swap kar leta hai:

$y = a^{x}$ ka feature	$x \leftrightarrow y$ swap	$y = lo g_{a} x$ ka feature
$(0, 1)$ se guzarta hai		== $(1, 0)$ == se guzarta hai
horizontal asymptote $y = 0$		vertical asymptote == $x = 0$ ==
saare $x$ ke liye defined		sirf $x > 0$ ke liye defined
range saare $y > 0$		range saare real $y$
$(1, a)$ se guzarta hai		$(a, 1)$ se guzarta hai

Figure — Graphs of logarithmic functions

Base ke hisaab se shape

Yeh itna dheere kyun badhta hai? $y$ ko $1$ badhane ke liye, aapko $x$ ko $a$ se multiply karna padta hai. Base $10$ ke saath $y = 6$ paane ke liye $x = 1 0^{6} = 1000000$ chahiye. Bada input → chota output. Yahi wajah hai ki logs "huge ranges compress" karne ke liye use hote hain (decibels, pH, Richter scale).

$lo g$ graphs ke transformations

Log laws use karke hum transformed graphs ko vertical shifts ki tarah likh sakte hain:

$lo g_{a} (k x) = lo g_{a} k + lo g_{a} x$

Toh andar ko constant $k$ se multiply karna bilkul waise hi hai jaise poore curve ko $lo g_{a} k$ upar shift karo. Yeh ek neat, testable fact hai.

$y = lo g_{a} x$ ke standard transformations:

$y = lo g_{a} x + c$ → $c$ upar shift karo.
$y = lo g_{a} (x - b)$ → $b$ daayein shift karo, asymptote $x = b$ par chali jaati hai.
$y = - lo g_{a} x$ → $x$ -axis mein reflect karo.

Worked examples

Common mistakes (steel-manned)

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

Logarithm ek "kitni baar multiply karta hoon" wali machine hai. $lo g_{10} 1000$ poochh raha hai "kitne 10s multiply karke 1000 banta hai?" → teen. Ab iska picture banao: graph left edge ke paas bahut neeche se shuru hota hai (tum kabhi $x = 0$ wali upar-neeche line ko touch nahi kar sakte), neeche $x = 1$ par cross karta hai (kyunki 1 ka $lo g$ hamesha 0 hota hai — 1 paane ke liye zero baar multiply karte ho), phir upar chadta hai lekin lazier aur lazier hota jaata hai, har extra step ke liye 10× zyaada input chahiye. Yeh literally exponential graph hai jo diagonal line ke saath rakhe mirror mein dekha ja raha hai.

Active recall

Kaun sa transformation

y = a^{x}

y = lo g_{a} x

par map karta hai?

Line

y = x

mein reflection (inverse function).

Har

y = lo g_{a} x

kis fixed point se guzarta hai?

(1, 0)

se, kyunki

lo g_{a} 1 = 0

y = lo g_{a} x

ka asymptote kya hai aur uska type kya hai?

x = 0

, ek vertical asymptote.

y = lo g_{a} x

(

a > 1

) ka domain aur range batao.

Domain

x > 0

; range saare real numbers.

y = lo g_{a} x

ka koi

y

-intercept kyun nahi hota?

Kyunki

x = 0

excluded hai (asymptote); 0 ka log undefined hai.

a > 1

ke liye,

lo g_{a} x

increasing hai ya decreasing, aur kitni speed se?

Increasing, lekin dheere —

y

ko 1 badhane ke liye

x

a

se multiply karna padta hai.

y = lo g_{a} x

par

(1, 0)

ke alawa hamesha kaun sa point hota hai?

(a, 1)

, kyunki

lo g_{a} a = 1

y = lo g_{a} (x - b)

ke graph ka description do.

y = lo g_{a} x

b

se daayein shift hua; asymptote

x = b

par.

lo g_{a} (a x)

ko ek shift ki tarah rewrite karo.

lo g_{a} x + 1

(1 upar shift),

lo g_{a} (a x) = lo g_{a} a + lo g_{a} x

use karke.

Graph reasoning se

lo g_{10} x = 3

solve karo.

x = 1 0^{3} = 1000

(

x = a^{y}

ki tarah rewrite karo).

Connections

Exponential functions and their graphs — reflection partner.
Inverse functions and reflection in y=x — shape ka geometric reason.
Laws of logarithms — inside-changes ko vertical shifts mein convert karo.
Natural logarithm ln and e — special base $a = e$ .
Solving equations with logarithms — graph se intersections padhna.
Transformations of graphs — shifts, reflections, stretches yahan apply hote hain.