Class 9 Maths Chapter 2 Logarithms Notes – Exercise 2.4 Solved (Free PDF Download)

Exercise 2.4 is where Logarithms starts to feel powerful. Exercise 2.3 taught you to find a log with Characteristic and Mantissa. Now, in Exercise 2.4, you learn the 4 Laws of Logarithms — four short rules that let you join logs together, split them apart, and solve sums no calculator can do in one step. The whole exercise runs on those four laws, plus the log table you already know. These notes cover Exercise 2.4 completely, in handwritten style and in very simple words. They are written for slow learners and for students whose first language is not English, so every step is short, every new word is explained the moment it appears, and every answer is checked.

Read This First: The 4 Laws in One Breath

Say this out loud twice: “Times is plus. Divide is minus. Power comes in front.” That is three of the four laws, and they do most of the work in this exercise. Law 1: log (a × b) = log a + log b — times becomes plus. Law 2: log (a / b) = log a − log b — divide becomes minus. Law 3: log aⁿ = n × log a — the small power jumps down in front. Law 4 is change of base: log_b x = log x / log b — you need it only in Question 2 (iii) and (iv). Two golden facts go with them, and you will use them again and again: log_a a = 1 (same number, same base, answer 1) and log_a 1 = 0 (log of 1 is always 0). Learn the four laws first, and every question in this exercise is just choosing which law to use.

Notes Details

Class	9 / Grade 9
Subject	Mathematics
Chapter	2 – Logarithms
Exercise	2.4 (complete)
Format	PDF (handwritten notes)
Pages	16
Language	Simple English
Price	Free

For International Students

These notes are labelled “Class 9”, the name used in Pakistan, India and Bangladesh — but the mathematics is the same everywhere. Class 9 in Pakistan is Grade 9 in the United States and Canada, and Year 10 in the United Kingdom, where logarithms appear in GCSE number and algebra topics. Students in the Gulf following a Pakistani, Indian or British curriculum school use the same chapter. The English here is deliberately simple: short sentences, everyday words, and no heavy textbook language, so the notes work just as well for a student whose first language is not English. They follow the PCTB (Punjab Curriculum and Textbook Board) Class 9 Mathematics book, Chapter 2 — Logarithms, and they are equally useful for other BISE boards, and for CBSE, ICSE and NCTB students studying the same logarithms topics. You can read the whole exercise on this page and download the free PDF — no sign-up, no payment, no email address.

What Is Included in These Notes?

The 4 Laws Toolbox (Pages 2–3)

All four laws in plain words, with the two golden facts — log_a a = 1 and log_a 1 = 0 — and the saying to learn by heart: “Times is plus. Divide is minus. Power comes in front.”

Question 1 — Find the Value, No Calculator (Pages 4–5)

All six parts, each solved by breaking big numbers into small parts and using the golden facts. Answers: 1, 7, −2, 2, 5 and 1.

Question 2 — Make ONE Log (Pages 6–7)

All six parts: join many logs into one single log. First push every number in front up as a power, then plus joins on top and minus goes at the bottom. Answers: log 45, log 27, 6 log_a b, log_3 x²y, log_5 (x⁴z / y) and ln (a²b³ / c⁴).

Question 3 — Open the Log Out, Expand (Pages 8–9)

Question 2 in reverse: one log becomes many small logs. Answers: log 11 − log 5, 3/2 log_5 2 + 3 log_5 a, 2 ln a + ln b − ln c, 1/9 (log x + log y − log z), 4/3 ln 2 + ln x, and 5 [log_2 (1 − a) − log_2 b].

Question 4 — Find x (Pages 10–11)

All six parts, with the key move: if log_b (thing) = n, then the thing = bⁿ — “the base jumps up.” Answers: 5, 4, −10, 5, 22 and 17/3.

Question 5 — Use the Log Table (Pages 12–13)

The same four steps every time: call the answer x, take log of both sides, put the table values in, then take antilog. Answers: 2.9601, 23.6194, 1.3388 and 14.2003.

Questions 6, 7 and 8 — Real-Life Sums (Pages 14–15)

An earthquake (M = 3), Abdullah’s money doubling in 14 years, and the temperature on a mountain (17.18 °C).

Check Your Answers (Page 16)

Every answer from all eight questions in one table, so you can check your own work in a minute.

Exercise 2.4 Detailed Walkthrough: Every Question Explained

Question 1 — Find the value, no calculator. The trick is always the same: break big numbers into small parts, then use the golden facts. For (i), log_2 18 − log_2 9, split 18 into 2 × 9, and the log_2 9 parts cut each other, leaving log_2 2 = 1. For (ii), log_2 64 + log_2 2, write 64 as 2⁶, so it becomes 6 + 1 = 7. For (iii), 1/3 log_3 8 − log_3 18, the 3 and the 1/3 cut, and the log_3 2 parts cut, leaving −2. For (iv), 2 log 2 + log 25, the power goes back up to give log 4 + log 25 = log 100 = 2. For (v), 1/3 log_4 64 + 2 log_5 25, both powers come in front to give 1 + 4 = 5. And for (vi), log_3 12 + log_3 0.25, the log_3 4 parts cut, leaving log_3 3 + log_3 1 = 1 + 0 = 1. The notes end this question with the habit to keep: when two logs are the same, they cut each other — look for the cut before you do any hard work.

Question 2 — Make one log. Push every number in front up as a power first, then join. For (i), 1/2 log 25 + 2 log 3 becomes log 5 + log 9, and plus joins them into log 45. For (ii), log 9 − log 1/3, minus means divide, and 9 ÷ 1/3 = 27, so the answer is log 27. For (iii), the two logs are multiplied together, so the powers come in front and then Law 4, change of base, makes the log 5 parts cut — leaving 6 log_a b. For (iv), 2 log_3 x + log_3 y becomes log_3 x²y. For (v), 4 log_5 x − log_5 y + log_5 z, the plus parts go on top and the minus part goes at the bottom: log_5 (x⁴z / y). For (vi), 2 ln a + 3 ln b − 4 ln c, ln is just a log with a special base, so the same laws give ln (a²b³ / c⁴). The easy rule to remember: plus = top, minus = bottom.

Question 3 — Expand. This is Question 2 in reverse, and the order matters: power first, then split. For (i), log (11 / 5) is a divide, so it becomes log 11 − log 5 — top first, bottom second. For (ii), log_5 √(8a⁶), the root means power 1/2, and 8 is 2³, so the answer is 3/2 log_5 2 + 3 log_5 a. For (iii), ln (a²b / c), the top splits with plus and the bottom goes with minus: 2 ln a + ln b − ln c. For (iv), log (xy / z)^(1/9), the power 1/9 comes in front first: 1/9 (log x + log y − log z). For (v), ln of the cube root of 16x³, the cube root means power 1/3, and 16 is 2⁴, so the answer is 4/3 ln 2 + ln x. For (vi), log_2 ((1 − a) / b)⁵, the power 5 comes in front, then divide becomes minus: 5 [log_2 (1 − a) − log_2 b]. And one careful warning from the notes: (1 − a) stays together in one bracket — do not split it.

Question 4 — Find x. The key move is one line: if log_b (thing) = n, then the thing = bⁿ. Say it like this: “The base jumps up.” For (i), log 2 + log x = 1 joins into log 2x = 1, the base 10 jumps up to give 10 = 2x, so x = 5. For (ii), log_2 x + log_2 8 = 5 joins into log_2 8x = 5, so 32 = 8x and x = 4. For (iii) and (iv), there is no log at all — both sides are made into powers of the same base, so the powers must be equal: (81)^x = (243)^(x+2) gives x = −10, and (1/27)^(x−6) = 27 gives x = 5. For (v), log (5x − 10) = 2, the base 10 jumps up to give 100 = 5x − 10, so x = 22. For (vi), log_2 (x + 1) − log_2 (x − 4) = 2, minus becomes one log of a divide, the base 2 jumps up to give (x + 1) / (x − 4) = 4, so x = 17/3.

Question 5 — Use the log table. The same four steps, every time: call the answer x, take log of both sides, put the table values in, then take antilog. For (i), (3.68 × 4.21) / 5.234, times becomes plus and divide becomes minus, so log x = 0.5658 + 0.6243 − 0.7188 = 0.4713, and antilog gives x = 2.9601. For (ii), 4.67 × 2.11 × 2.397, all times means all plus, so log x = 0.6693 + 0.3243 + 0.3797 = 1.3733, and antilog gives x = 23.6194. For (iii), the power 2 comes in front before the table values go in, giving log x = 0.1267 and x = 1.3388. For (iv), the cube root is power 1/3, giving log x = 1.1523 and x = 14.2003. The reminder at the end is worth learning: log takes a number in, antilog brings the number out.

Questions 6, 7 and 8 — Real life. Question 6 is an earthquake: M = log_10 (A / A₀), with A = 10,000 and A₀ = 10, so M = log_10 1000 = 3. Question 7 is Abdullah’s money: Rs. 100,000 growing by 5% a year doubles when 2 = (1.05)^t, so log 2 = t × log 1.05, and t = 0.3010 / 0.0212 — about 14 years. Question 8 is mountain temperature: T = 20 × (0.97)⁵, so log T = log 20 + 5 log 0.97 = 1.235, and antilog gives T = 17.18 °C. Three different stories, one method: take log of both sides, use the laws, then antilog.

Frequently Asked Questions

What are the 4 laws of logarithms, in simple words?

Times becomes plus, divide becomes minus, power comes in front, and change of base lets you swap one base for another. With those four and the two golden facts — log_a a = 1 and log_a 1 = 0 — you can solve every sum in this exercise.

What is “change of base” and when do I use it?

It is Law 4: log_b x = log x / log b. You use it when two logs have different bases and you need them to match — in this exercise, only in Question 2 (iii) and (iv).

What is ln?

It is just a log with a special base, the number e. The same four laws work for ln exactly as they do for log — Question 2 (vi) and Question 3 (iii) and (v) show it.

When do I use the log table and when do I use the laws?

Use the laws when the numbers are powers of the same base or break into small parts — that is Questions 1 to 4. Use the log table when the numbers are ordinary decimals like 3.68 and 4.21 — that is Question 5 and the real-life sums.

Which syllabus do these notes follow?

They follow the PCTB (Punjab Curriculum and Textbook Board) Class 9 Mathematics book, Chapter 2 — Logarithms, and they are equally useful for other BISE boards teaching the same chapter, and for international students studying logarithms at Grade 9 / Year 10 level.

Can I print these notes or share them with my class fellows?

Yes. The PDF prints cleanly on A4 paper, and you may share it freely with your class fellows and study groups.

Conclusion

Exercise 2.4 gives you the working rules of logarithms: four laws, two golden facts, and the log table for the numbers the laws cannot reach — then three real-life sums to show why it matters. Download the free PDF, practise until “times is plus, divide is minus, power comes in front” is automatic, and Chapter 2 will feel finished, not just studied. For the Characteristic and Mantissa tools that came before, see Exercise 2.3, and find everything on our Free Class 9 Notes and Free Math Notes pages — everything on AllFreeAssets.com is free, always.

Download Class 9 Maths Logarithms Notes PDF

PDF • 16 Pages • Class 9 Maths • Exercise 2.4