3.2.8
Ek logarithm ek sawaal poochta hai : "log b x "b x
Kyunki logs disguise mein exponents hain , exponents ka har rule logs ka rule ban jaata hai.
Numbers ko multiply karna ⇄ unke exponents ko add karna. Divide karna ⇄ exponents ko subtract karna. Power mein raise karna ⇄ exponent ko multiply karna. Yahi ek mirror hai jo is pura topic hai.
Definition Log ki definitionBase b > 0 , b = 1 x > 0 log b x = y ⟺ b y = x b l o g b x = x and log b ( b k ) = k
Restrictions kyun? Hume b > 0 b = 1 1 y = 1 x > 0 hamesha positive hota hai — real numbers mein aap kabhi 0
Har baar ka trick: logs ko exponents ke naam do , exponent laws use karo, phir wapas translate karo.
log b ( M N ) = log b M + log b N Step 1. Maano p = log b M q = log b N Kyun? Unknown logs ko naam do taaki unhe exponents ki tarah use kar sakein.
Step 2. Definition se, b p = M b q = N Kyun? Yahi "log matlab hai — log ko undo karke exponential form mein laao.
Step 3. Multiply karo: M N = b p ⋅ b q = b p + q Kyun? Same-base multiplication exponents add karta hai (b p b q = b p + q
Step 4. Dono sides ka log b log b ( M N ) = log b ( b p + q ) = p + q Kyun? Kyunki log b ( b k ) = k
Step 5. p , q log b ( M N ) = log b M + log b N . ■
log b ( M / N ) = log b M − log b N Step 1–2. Same shuruat: p = log b M , q = log b N b p = M , b q = N Step 3. Divide karo: N M = b q b p = b p − q Kyun? Same-base division exponents subtract karta hai.
Step 4. log b ( N M ) = log b ( b p − q ) = p − q Step 5. = log b M − log b N . ■
log b ( M k ) = k log b M Step 1. Maano p = log b M b p = M Step 2. Dono sides ko power k M k = ( b p ) k = b p k Kyun? Power of a power exponents multiply karta hai (( b p ) k = b p k Step 3. log b ( M k ) = log b ( b p k ) = p k = k p Step 4. = k log b M . ■
Bonus (reciprocal): k = − 1 log b ( 1/ M ) = − log b M
Intuition Jo 90% marks dilata hai
Definition ⇄ exponential form (har log ko b something Multiply→add, divide→subtract, power→multiply-out-front. Expressions ko fluently combine aur split karo — yahi exam questions test karte hain.
Worked example Ek log mein combine karo2 log 3 x + log 3 5 − log 3 y Step 1. 2 log 3 x = log 3 x 2 Kyun? Power rule ulta — coefficient ko power ke roop mein upar le jao.
Step 2. log 3 x 2 + log 3 5 = log 3 ( 5 x 2 ) Kyun? Logs ka sum → product.
Step 3. log 3 ( 5 x 2 ) − log 3 y = log 3 ( y 5 x 2 ) . Kyun? Difference → quotient. ✅
Worked example Bina calculator ke evaluate karolog 2 48 − log 2 3 Step 1. Quotient rule: = log 2 ( 3 48 ) = log 2 16 Kyun? Difference ek aisi division mein collapse ho jaata hai jise hum simplify kar sakte hain.
Step 2. 16 = 2 4 log 2 16 = 4 Kyun? log b ( b k ) = k
Worked example Equation solve karolog 5 ( x ) + log 5 ( x − 4 ) = 1 Step 1. Product rule: log 5 ( x ( x − 4 ) ) = 1 Step 2. Rewrite karo: x ( x − 4 ) = 5 1 = 5 Kyun? log 5 A = 1 ⇔ A = 5 Step 3. x 2 − 4 x − 5 = 0 ⇒ ( x − 5 ) ( x + 1 ) = 0 ⇒ x = 5 x = − 1 Step 4. x = − 1 log 5 ( x ) Answer x = 5 Kyun? Domain: arguments > 0
log ( M + N ) = log M + log N Kyun sahi lagta hai: product rule ek distributive law jaisi dikhti hai, to log log karte hain + Fix: logs products ko sums mein badlte hain, sums ko sums mein nahi. log b ( M N ) = log b M + log b N log b ( M + N ) simplify nahi ho sakta . Test karo: log 10 ( 1 + 1 ) = log 10 2 ≈ 0.30 log 10 1 + log 10 1 = 0
log N log M = log M − log N Kyun sahi lagta hai: log ke andar quotient subtraction ban jaata hai, to students logs ke ratio ke liye bhi subtract kar dete hain.
Fix: subtraction sirf log ( M / N ) change-of-base result hai log N log M = log N M
( log M ) 2 = 2 log M Kyun sahi lagta hai: power rule exponents ko neeche laata hai — lekin sirf woh exponents jo argument M hain.
Fix: log ( M 2 ) = 2 log M ( log M ) 2 = log M ⋅ log M pura log squared) — koi simplification nahi.
Common mistake Solve karte waqt domain bhool jaanaKyun sahi lagta hai: do algebraic solutions milte hain aur aap dono chahte ho.
Fix: hamesha check karo ki har answer mein har logged argument > 0
Recall Product rule prove karo (note dhako, khud karo)Maano p = log b M , q = log b N b p = M , b q = N M N = b p + q log b ( M N ) = p + q = log b M + log b N .
Recall Log ke andar kaunsa operation subtraction ban jaata hai?Division: log b ( M / N ) = log b M − log b N
Recall 12 saal ke bache ko samjhao (Feynman)Log ek "kitni baar multiply kiya?" counter hai. Agar aap do cheezein ek saath multiply karo, to bas add karo ki har cheez ke liye kitni multiplications lagein — to logs "times" ko "plus" mein badal dete hain. Divide karna matlab kuch wapas lena, to logs "divide" ko "minus" mein badal dete hain. Aur "M 3 M M 3 guna hai — isliye power saamne aa jaata hai.
Mnemonic Yaad rakho kaunsa law kaunsa hai"Multiply, Add — Divide, Subtract — Power, Ping-out-front."
Ya: logs peace-maker hain — woh mushkil operation ko usse neeche wale aasaan operation mein badal dete hain (×→+, ÷→−, power→×).
log b x = y b y = x b > 0 , b = 1 , x > 0
Logs ka product rule batao. log b ( M N ) = log b M + log b N
Logs ka quotient rule batao. log b ( M / N ) = log b M − log b N
Logs ka power rule batao. log b ( M k ) = k log b M k
Product rule prove karne ka key step kya hai? Same-base multiplication exponents add karta hai: b p b q = b p + q
Power rule prove karne ka key step kya hai? Power of a power exponents multiply karta hai: ( b p ) k = b p k
log 2 48 − log 2 3 log 2 16 = 4
Kya log ( M + N ) = log M + log N Nahi — logs products ko sums mein badlte hain, sums ko sums mein nahi.
log N log M log N M log M − log N
Log equations solve karte waqt kuch solutions reject kyun karte hain? Har logged argument > 0
log b ( 1/ M ) − log b M k = − 1
log b x x > 0 Positive base ko kisi bhi real power par raise karo to result hamesha positive hota hai, isliye non-positive x
Definition log_b x=y iff b^y=x
b^(log x)=x and log_b b^k=k
Restrictions b>0, b!=1, x>0
Product rule log MN = log M + log N
Quotient rule log M/N = log M - log N
Power rule log M^k = k log M
