Solving exponential equations using logarithms

3.2.10 · Maths › Exponentials & Logarithms

Problem kya hai?

HUMEIN KYA CHAHIYE: ek key fact — logarithms ki power law.

Common mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke bache ko samjhao

Socho ek number seedhi ke upar (exponent mein) chhupta hai, aur tum tak pahunch nahi sakte. Logarithm ek special elevator ki tarah hai jo upar jo bhi hai use seedha neeche floor pe le aata hai, aur use ek normal number mein badal deta hai jise tum aasaan arithmetic se idhar-udhar kar sako. Toh "2 ki power kya equals 7?" solve karne ke liye, tum log elevator bhejte ho, kya ko neeche laate ho, aur phir bas divide karte ho.

80/20 — zaroori 20%

Sirf ek law heavy lifting karti hai: $lo g (m^{k}) = k lo g m$ .
Pehle exponential ko isolate karo, phir log lo.
Result hamesha $x = \frac{lo g c}{lo g b}$ hota hai — ek ratio, kabhi difference nahi.
Agar dono sides ka base same ho, toh exponents match karo (cleaner, exact).

Flashcards

Kaun sa operation ek exponential ko "undo" karta hai, jisse tum ek phanse hue exponent ke liye solve kar sako?

Dono sides ka logarithm lena (log aur exponential inverse operations hain).

Logarithms ki power law batao.

lo g_{b} (m^{k}) = k lo g_{b} (m)

— exponent ek multiplier ban jaata hai.

lo g_{b} (m^{k}) = k lo g_{b} m

kyun hota hai, derive karo.

Maano

y = lo g_{b} m

isliye

b^{y} = m

;

k

pe raise karo:

b^{y k} = m^{k}

; log form mein

lo g_{b} (m^{k}) = y k = k lo g_{b} m

2^{x} = 7

solve karo (method + answer).

x lo g 2 = lo g 7

, isliye

x = lo g 7/ lo g 2 \approx 2.807

5 \cdot 3^{x} = 60

ke liye, log lene se PEHLE kya karna hai?

Power ko isolate karne ke liye 5 se divide karo:

3^{x} = 12

lo g c / lo g b

ki jagah

lo g (c / b)

kyun nahi likh sakte?

lo g c / lo g b

ek ratio hai (=

lo g_{b} c

);

lo g (c / b) = lo g c - lo g b

ek difference hai — alag values hain.

2^{x + 1} = 8^{x}

ko logs ke bina solve karo.

8 = 2^{3}

, isliye

x + 1 = 3 x

, jo

x = \frac{1}{2}

deta hai.

7^{2 x - 1} = 40

mein, power law apply karne pe kya neeche aata hai?

Poora exponent

(2 x - 1)

bracket ke saath:

(2 x - 1) lo g 7 = lo g 40

b^{x} = c

ka general solution kya hai?

x = \frac{lo g c}{lo g b} = lo g_{b} c

(koi bhi log base kaam karti hai).

Connections

Laws of Logarithms — yahan use ki gayi power/product/quotient rules.
Change of Base Formula — $lo g_{b} c = \frac{l o g c}{l o g b}$ exactly hamaara result hai.
Inverse Functions — kyun log, exp ko undo karta hai.
The Number e and Natural Logarithms — $lo g_{10}$ ki jagah $ln$ use karna.
Exponential Growth and Decay — yeh equations kahan appear hoti hain (half-life, interest).
Solving Equations Graphically — $y = b^{x}$ aur $y = c$ ka intersection view.

Solving exponential equations using logarithms

Problem kya hai?

Universal method

Worked examples