Logarithmलघुगणक questions and answersप्रश्न और उत्तर

How logarithm worksलघुगणक को समझिए

Because powers of the same base add when the numbers multiply, logs of a product add: log (xy) = log x + log y. In the same way the log of a quotient is a difference, and a power inside the log comes out in front as a multiplier: log x³ = 3 log x.

The change-of-base rule log_b x = log x / log b lets you work in any base you like. It explains why log₄ 32 = 5/2 (32 = 2⁵ and 4 = 2²) and why a chain such as log₂ 7 × log₇ 32 collapses to log₂ 32 = 5: the middle logs cancel.

In an equation, first combine the logs, then rewrite without them: log₂ x + log₂ (x + 6) = 4 becomes x(x + 6) = 16. Finally check every root, because a log is defined only for positive numbers; here x = −8 is rejected and x = 2 is the answer.

Each trick says when to use it, why it works and when it fails. Formula tricks are checked by code over their whole range.हर ट्रिक बताती है कि उसे कब इस्तेमाल करें, वह क्यों काम करती है और कब नहीं। सूत्र वाली ट्रिक्स को कोड से उनकी पूरी रेंज पर जाँचा गया है।

Answers and full solutions are shown. Hide them to test yourself first.उत्तर और पूरे हल दिखाए गए हैं। पहले खुद को परखने के लिए इन्हें छिपाइए।

1. Example 1उदाहरण 1 · Laws of logarithms · Easyआसान

   Find the value of log₆ 216 + log₆ 1,296 + log₆ 6.

   1. 8
   2. 9
   3. 12
   4. 16,79,616
   Answer:उत्तर: 8

   1. 216 = 6³, 1,296 = 6⁴, 6 = 6¹.
   2. So the expression is 3 + 4 + 1 = 8.

2. Example 2उदाहरण 2 · Laws of logarithms · Mediumमध्यम

   If log₃ a = 4 and log₃ b = 5, find log₃ (ab).

   1. 2
   2. 9
   3. 10
   4. 20
   Answer:उत्तर: 9

   1. log (ab) = 1 log a + 1 log b.
   2. = 1 × 4 + 1 × 5 = 9.

3. Example 3उदाहरण 3 · Laws of logarithms · Hardकठिन

   Solve for x: log₂ x + log₂ (x − 12) = 6.

   1. −4
   2. 4
   3. 16
   4. 38
   Answer:उत्तर: 16

   1. log₂ [x(x − 12)] = 6, so x(x − 12) = 64.
   2. x² − 12x − 64 = 0 ⇒ (x − 16)(x + 4) = 0.
   3. x = 16 or x = −4; logs need x > 0 and x − 12 > 0, so x = 16.

4. Example 4उदाहरण 4 · Laws of logarithms · Mediumमध्यम

   Find the value of log to the base 25 of √25.

   1. 1/2
   2. 3/4
   3. 1
   4. 2
   Answer:उत्तर: 1/2

   1. 25 = 5², √25 = 5^(2/2).
   2. Let the value be x: (5²)ˣ = 5^(2/2), so 2 × x = 1.
   3. x = 1 ÷ 2 = 1/2.

5. Example 5उदाहरण 5 · Laws of logarithms · Mediumमध्यम

   If log₃ a = 7 and log₃ b = 7, find log₃ (a²b²).

   1. 14
   2. 28
   3. 98
   4. 196
   Answer:उत्तर: 28

   1. log (a²b²) = 2 log a + 2 log b.
   2. = 2 × 7 + 2 × 7 = 28.

6. Example 6उदाहरण 6 · Laws of logarithms · Mediumमध्यम

   Find the value of log to the base 8 of √128.

   1. 6/7
   2. 7/6
   3. 7/2
   4. 21/2
   Answer:उत्तर: 7/6

   1. 8 = 2³, √128 = 2^(7/2).
   2. Let the value be x: (2³)ˣ = 2^(7/2), so 3 × x = 7/2.
   3. x = 7/2 ÷ 3 = 7/6.

Common mistakesआम गलतियाँ

• Writing log (a + b) = log a + log b; the product rule is log (ab) = log a + log b.
• Writing log xⁿ as (log x)ⁿ instead of n log x.
• Keeping a root of a log equation that makes some log argument zero or negative.

Frequently asked questionsअक्सर पूछे जाने वाले प्रश्न