3.2.11
Ek logarithmic equation mein unknown andar chhupa hota hai log ke, jaise log 2 ( x ) = 5 logarithm ko undo karna hoga. Sabse powerful idea yeh hai:
log b ( a ) = c ⟺ b c = a ek sawaal poochta hai : "log 2 8 "2 ko kis power par raise karein ki 8 mile?" Log equations solve karna = us sawaal ko uske exponential answer mein rewrite karna, ya bahut saare logs ko ek mein compress karna taaki hum unhe undo kar sakein.
Definition Logarithmic equationEk aisi equation jisme unknown, logarithm ke argument mein (ya kabhi kabhi base mein) aata hai, jaise log 3 ( 2 x − 1 ) = 4 x dono sides defined aur equal hoon.
Zaroori restriction (domain): log b ( N )
argument N > 0 0
base b > 0 b = 1
Isliye har solution ko N > 0
Kyun kaam karta hai: dono sides par base raise karne se log "cancel" ho jaata hai kyunki b l o g b N = N
log b ( N ) = c ⟹ b l o g b N = b c ⟹ N = b c
Kyun: log function one-to-one hota hai (strictly increasing/decreasing), isliye
log b M = log b N ⟹ M = N ( M , N > 0 ) .
Worked example Example 1 — single logSolve karo log 2 ( 3 x − 1 ) = 4
Step 1. Argument already left side par ek log ke roop mein isolated hai. Kyun? Taaki hum seedha exponentiate kar sakein.
Step 2. Exponential form mein rewrite karo: 3 x − 1 = 2 4 = 16 Kyun? log b N = c ⟺ N = b c Step 3. Solve karo: 3 x = 17 ⇒ x = 3 17 Step 4 (domain check). Argument = 3 ( 3 17 ) − 1 = 16 > 0 Kyun? Log ka argument positive hona chahiye.
x = 3 17
Worked example Example 2 — do logs combine karoSolve karo log 5 ( x ) + log 5 ( x − 4 ) = 1
Step 1. Product law use karke combine karo: log 5 ( x ( x − 4 ) ) = 1 Kyun? Do alag logs undo nahi ho sakte; ek log ho sakta hai.
Step 2. Exponentiate karo: x ( x − 4 ) = 5 1 = 5 Kyun? log 5 N = 1 ⟺ N = 5 Step 3. Expand karo: x 2 − 4 x − 5 = 0 ⇒ ( x − 5 ) ( x + 1 ) = 0 ⇒ x = 5 x = − 1 Step 4 (check!). Chahiye x > 0 aur x − 4 > 0 x > 4
x = 5 5 > 0 1 > 0 x = − 1 log 5 ( − 1 ) reject .
x = 5
Worked example Example 3 — dono sides par logsSolve karo log 3 ( 2 x + 1 ) = log 3 ( x + 7 )
Step 1. Dono sides same base → arguments equal karo: 2 x + 1 = x + 7 Kyun? log Step 2. x = 6 Step 3 (check). 2 ( 6 ) + 1 = 13 > 0 6 + 7 = 13 > 0 x = 6
Worked example Example 4 — log mein chhupa quadraticSolve karo ( log 2 x ) 2 − 3 log 2 x + 2 = 0
Step 1. Maano u = log 2 x Kyun? Equation x log mein quadratic hai.
Step 2. u 2 − 3 u + 2 = 0 ⇒ ( u − 1 ) ( u − 2 ) = 0 ⇒ u = 1 u = 2 Step 3. Back-substitute karo: log 2 x = 1 ⇒ x = 2 log 2 x = 2 ⇒ x = 4 Step 4 (check). Dono 2 > 0 , 4 > 0 x = 2 or x = 4
Common mistake Domain check karna bhool jaanaGalat sahi lagta hai kyunki: algebra ne ek valid number diya, toh surely yeh solution hai. Lekin original equation wahan undefined ho sakti hai.
Fix: Solve karne ke baad, substitute karke confirm karo ki har log argument > 0
log ( A ) + log ( B ) = log ( A + B ) Sahi lagta hai kyunki: yeh symmetric aur "distributive" lagta hai. Lekin logs products ko sums mein badalta hai, sums ko sums mein nahi.
Fix: log A + log B = log ( A B ) log ( A + B ) koi rule nahi hai.
Common mistake Ek log jo number ke equal hai use "arguments equate" karke cancel karnalog 5 x + log 5 ( x − 4 ) = 1 x + ( x − 4 ) = 1 Sahi lagta hai kyunki: aap tab equate kar sakte ho jab dono sides single logs hon.
Fix: Right side ek number hai, log nahi. Pehle combine karo, phir exponentiate karo (= 5 1
Recall Feynman: ek 12-saal ke bacche ko samjhaoLogarithm ek "power detective" hai. log 2 8 "kitne 2 multiply karun taki 8 mile?" — answer 3 hai. Toh log equation ek paheli hai jisme detective ke sawaal mein ek number missing hai. Isko solve karne ke liye tum paheli ko ulta karte ho: "kaunsi power yeh number deti hai?" poochne ki jagah tum kehte ho "number = base ko us power par raise karo," aur ab normal algebra ho jaati hai. Golden rule: end mein apna answer wapas daalo — kyunki positive base se tum kabhi nahi pooch sakte "kaunsi power negative number deti hai," isliye jo bhi answer argument ko negative banata hai woh ek trick hai aur bahar ho jaata hai.
Mnemonic Workflow yaad karoC.E.C. → C ombine, E xponentiate, C heck.
("Cats Eat Cheese " — hamesha Check karke khatam karo domain, warna billi bhookhi rah jaati hai.)
log b N = c N = b c
Log equation ke solutions hamesha check kyun karne chahiye? Kyunki log arguments > 0 ≤ 0
log a M + log a N = ? log a ( M N )
log a M − log a N = ? log a ( M / N )
k log a M = ? log a ( M k )
Arguments directly kab equal kar sakte ho? Jab dono sides same base ke single logs hon: log b M = log b N ⇒ M = N
Solve karo log 2 ( 3 x − 1 ) = 4 3 x − 1 = 16 ⇒ x = 17/3
b l o g b N = N Kyunki log b b ( ⋅ )
Kya log ( A ) + log ( B ) = log ( A + B ) Nahi — yeh log ( A B ) log ( A + B )
( log x ) 2 − 3 log x + 2 = 0 u = log x u 2 − 3 u + 2 = 0
Log equation: unknown in argument
Domain: argument N>0, base b>0 not 1
Strategy 1: Isolate and exponentiate
Strategy 2: Combine logs then exponentiate
Strategy 3: Same base, equate arguments
Index laws b^x b^y = b^x+y
Log laws: product, quotient, power