Log 6: Natural Log Vs. Common Log Explained

Namaste! Aapse sawal pucha gaya hai ki log 6 ka maan kya hota hai, natural log (ln) aur common log (log₁₀) dono ka istemal karke. Chaliye, is sawaal ka ek dam vistaar se aur sahi jawab jante hain.

Correct Answer

Natural log ke liye, log 6 ka maan lagbhag 1.7917 hota hai, jabki common log ke liye, log 6 ka maan lagbhag 0.7781 hota hai.

Detailed Explanation

Logarithms (log) mathematical functions hote hain jo yeh batate hain ki kisi number ko base (aadhaar) ki kitni power tak badhaya jaye ki wo number mil jaye. Jaise, agar $2^3 = 8$ hai, toh $log_2 8 = 3$ hoga. Yahan 2 base hai, 8 number hai, aur 3 logarithm ka maan hai.

Jab hum "log" likhte hain bina koi base mention kiye, toh context ke hisab se base alag-alag ho sakta hai. Lekin jab hum "natural log" ya "common log" ki baat karte hain, toh base nishchit hota hai.

Key Concepts:

• Common Logarithm (log₁₀): Iska base 10 hota hai. Jab bhi aapko sirf "log" likha dikhe, khaas kar ke science aur engineering mein, toh aksar iska matlab common logarithm hi hota hai. Yeh numbers ko scientific notation mein represent karne ke liye bahut upyogi hota hai.
• Natural Logarithm (ln): Iska base e hota hai, jahan 'e' Euler's number hai, jiska maan lagbhag 2.71828 hota hai. Natural logarithm calculus, economics, aur physics jaise fields mein bahut common hai, kyunki yeh growth aur decay processes ko model karne mein madad karta hai.
• Change of Base Formula: Ek common logarithm ka maan dusre base wale logarithm ke maan mein badalne ke liye yeh formula istemal hota hai: $log_b a = \frac{log_c a}{log_c b}$. Yahan hum base 'c' ka istemal karke base 'b' wale logarithm ko nikal rahe hain.

Calculating log 6 using Common Logarithm (log₁₀):

Jab hum log 6 ka maan common logarithm (base 10) se nikalte hain, toh hum yeh pooch rahe hain ki "10 ko kitni power tak badhayein ki 6 mil jaye?"

Mathematically, hum $log_{10} 6$ ka maan nikal rahe hain.

Iska maan nikalne ke liye hum properties of logarithms ka upyog kar sakte hain, jaise ki $log(ab) = log a + log b$. Is case mein, hum 6 ko $2 \times 3$ ke roop mein likh sakte hain:

$log_{10} 6 = log_{10} (2 \times 3)$

Ab logarithm ki property lagane par:

$log_{10} 6 = log_{10} 2 + log_{10} 3$

Hum jante hain ki:

• $log_{10} 2 \approx 0.3010$
• $log_{10} 3 \approx 0.4771$

In dono maanon ko jodne par:

$log_{10} 6 \approx 0.3010 + 0.4771$

$log_{10} 6 \approx 0.7781$

So, common logarithm ke hisab se log 6 ka maan lagbhag 0.7781 hai.

Calculating log 6 using Natural Logarithm (ln):

Natural logarithm mein base 'e' hota hai. Toh hum $ln 6$ ya $log_e 6$ ka maan nikal rahe hain. Iska matlab hai ki "e (lagbhag 2.71828) ko kitni power tak badhayein ki 6 mil jaye?"

Hum natural logarithm ke liye bhi properties of logarithms ka upyog kar sakte hain:

$ln 6 = ln (2 \times 3)$

$ln 6 = ln 2 + ln 3$

Hum jante hain ki:

• $ln 2 \approx 0.6931$
• $ln 3 \approx 1.0986$

In dono maanon ko jodne par:

$ln 6 \approx 0.6931 + 1.0986$

$ln 6 \approx 1.7917$

So, natural logarithm ke hisab se log 6 ka maan lagbhag 1.7917 hai.

Converting Between Bases (Change of Base Formula):

Hum change of base formula ka upyog karke bhi ek logarithm ko dusre mein badal sakte hain. Maan lijiye humein $log_{10} 6$ ka maan pata hai (0.7781) aur humein $ln 6$ ka maan nikalna hai. Hum jante hain ki $ln 6 = log_e 6$. Change of base formula ke anusar:

$log_e 6 = \frac{log_{10} 6}{log_{10} e}$

Humein $log_{10} e$ ka maan pata hona chahiye, jo lagbhag 0.4343 hota hai.

$ln 6 \approx \frac{0.7781}{0.4343}$

$ln 6 \approx 1.7917$

Isi tarah, agar humein $ln 6$ ka maan pata hai (1.7917) aur $log_{10} 6$ ka maan nikalna hai:

$log_{10} 6 = \frac{ln 6}{ln 10}$

Humein $ln 10$ ka maan pata hona chahiye, jo lagbhag 2.3026 hota hai.

$log_{10} 6 \approx \frac{1.7917}{2.3026}$

$log_{10} 6 \approx 0.7781$

Yeh dikhata hai ki dono tarikon se nikalne wale maan sahi hain aur ek dusre se related hain.

When to Use Which Logarithm?

• Common Logarithm ($log_{10}$): Yeh un numbers ke liye istemal hota hai jo powers of 10 mein aasan tarike se represent ho sakte hain. Jaise scientific notation mein. Yeh bahut sari purani calculation tools (jaise slide rules) mein bhi use hota tha.
• Natural Logarithm ($ln$): Yeh exponential growth and decay, compounding interest, aur calculus mein derivative aur integrals mein bahut use hota hai. Kai scientific formulas mein 'e' ka presence hone ke karan, natural logarithm bahut practical ho jata hai.

Log 6 ka maan dono hi tareekon se alag hai kyunki unke base alag hain. Dono hi maan sahi hain, bas alag-alag reference point (base) ke liye hain.

Conclusion

• Logarithms kisi number ko uske base ki power ke roop mein represent karne ka ek tarika hai.
• Common Logarithm ka base 10 hota hai, aur $log_{10} 6 \approx 0.7781$.
• Natural Logarithm ka base 'e' (Euler's number) hota hai, aur $ln 6 \approx 1.7917$.
• Logarithm ke values base par nirbhar karte hain.
• Change of Base formula ka upyog karke ek base wale logarithm ko dusre base wale logarithm mein badla ja sakta hai.

Asha hai yeh vistaar se samjhane mein aapki madad karega! Agar koi aur sawal ho toh zarur puchiye. Dhanyawad!