Review Exercise 2 is the whole of Chapter 2 in one exercise. Everything from Exercises 2.1 to 2.4 comes back here: scientific notation, log form and power form, the laws of logarithms, the log table, and one real-life population sum. If you can do this review, the chapter is yours. These notes cover Review Exercise 2 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: Your Toolbox
Every question in this review uses the same small rules. Scientific notation: write the number as a × 10ⁿ, where a is from 1 to 9.99. Point moves right for small numbers, so the power is minus. Point moves left for big numbers, so the power is plus. Log form and power form are the same thing: log_b N = p means b^p = N. Say it: “The base jumps up.” The laws of logarithms: times is plus, log (a × b) = log a + log b; divide is minus, log (a / b) = log a − log b; power comes in front, log aⁿ = n × log a. And the two golden facts: log_a a = 1 and log_a 1 = 0. And remember: log = Characteristic + Mantissa. The Characteristic is the whole-number part. The Mantissa is the decimal part.
Notes Details
| Class | 9 / Grade 9 |
| Subject | Mathematics |
| Chapter | 2 – Logarithms |
| Exercise | Review Exercise 2 (complete) |
| Format | PDF (handwritten notes) |
| Pages | 14 |
| 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 Toolbox (Pages 1–2)
Scientific notation, the log form and power form flip, the three laws, the two golden facts, and log = Characteristic + Mantissa — everything the review needs, on two pages.
Question 1 — MCQs (Pages 2–3)
All ten multiple-choice questions, each with the reason for the answer. Answers: (i) C, (ii) B, (iii) B, (iv) D, (v) A, (vi) C, (vii) D, (viii) C, (ix) D, (x) C.
Question 2 — Scientific Notation (Page 4)
Three parts: 0.000567 = 5.67 × 10⁻⁴, 734 = 7.34 × 10², and 0.33 × 10³ = 3.3 × 10².
Questions 3 and 4 — Ordinary Notation and Log Form (Page 5)
Back to ordinary numbers: 2600, 0.0008794 and 0.000006. Then power form into log form: log₃2187 = 7, logₐc = b and log₁₂144 = 2.
Questions 5 and 6 — Exponential Form and Find x (Pages 6–7)
Log form back into power form, then find x: 3, −1/2 and −3/5.
Question 7 — Make ONE Log (Page 8)
Three parts: log (x⁷ / y⁶), log 2 and log₅2.
Question 8 — Open the Log Out, Expand (Page 9)
Three parts: log x + log y + 6 log z, (1/6)(5 log₃m + 3 log₃n), and 3/2 (log 2 + log x).
Question 9 — Use the Log Table (Pages 10–11)
Three sums: ∛68.24 = 4.086, 319.8 × 3.543 = 1133.182, and (36.12 × 750.9) / (113.2 × 9.98) = 24.01.
Question 10 — Population of a City (Page 12)
A city of 22 million in 2016 growing at 2.5% a year reaches 35 million in the year 2035.
Check Your Answers (Pages 13–14)
Every answer from all ten questions in one table, so you can check your own work in a minute.
Review Exercise 2 Detailed Walkthrough: Every Question Explained
Question 1 — MCQs. Four options are given, pick the correct one, and the reason is under each question. (i) The standard form of 5.2 × 10⁶ is 5,200,000 — (c) — move the point 6 places right. (ii) Scientific notation of 0.00034 is 3.4 × 10⁻⁴ — (b) — the point moves 4 places right, so the power is minus 4. (iii) The base of the common logarithm is 10 — (b). (iv) log₂2³ = 3 — (d) — the power comes in front, 3 × log₂2, and log₂2 = 1. (v) log 100 = 2 — (a) — because 10² = 100. (vi) If log 2 = 0.3010, then log 200 = 2.3010 — (c) — because 200 = 2 × 100, so log 200 = log 2 + log 100 = 0.3010 + 2. (vii) log (0) is undefined — (d) — no power of 10 can make 0. (viii) log 10,000 = 4 — (c) — because 10⁴ = 10,000. (ix) log 5 + log 3 = log 15 — (d) — times is plus. (x) 3⁴ = 81 in logarithmic form is log₃81 = 4 — (c) — the base stays the base, the power becomes the answer.
Question 2 — Scientific notation. Count the places the point moves. For (i), 0.000567, the point moves 4 places right, so the power is minus: 5.67 × 10⁻⁴. For (ii), 734, the point moves 2 places left, so the power is plus: 7.34 × 10². For (iii), 0.33 × 10³, first write 0.33 as 3.3 × 10⁻¹, then add the powers: 3.3 × 10².
Question 3 — Ordinary notation. The power tells how far to move the point. For (i), 2.6 × 10³, move the point 3 places right: 2600. For (ii), 8.794 × 10⁻⁴, move the point 4 places left: 0.0008794. For (iii), 6 × 10⁻⁶, move the point 6 places left from 6: 0.000006.
Question 4 — Log form. The base stays the base, the power becomes the answer. For (i), 3⁷ = 2187 becomes log₃2187 = 7. For (ii), aᵇ = c becomes logₐc = b. For (iii), 12² = 144 becomes log₁₂144 = 2.
Question 5 — Exponential form. This is Question 4 in reverse: the base jumps up. For (i), log₄8 = x becomes 4ˣ = 8. For (ii), log₉729 = 3 becomes 9³ = 729. For (iii), log₄1024 = 5 becomes 4⁵ = 1024.
Question 6 — Find x. For (i), log₉x = 0.5, the base 9 jumps up: x = 9^0.5, and the square root of 9 is 3, so x = 3. For (ii) and (iii), both sides are made into powers of the same base, so the powers must be equal: x = −1/2 and x = −3/5.
Question 7 — Make one log. Push every number in front up as a power first, then join. For (i), 7 log x − 6 log y becomes log (x⁷ / y⁶) — plus goes on top, minus goes at the bottom. For (ii), the logs join into log 2. For (iii), change of base makes the parts cut, leaving log₅2.
Question 8 — Expand. This is Question 7 in reverse: power first, then split. For (i), log (xyz⁶) becomes log x + log y + 6 log z. For (ii), the power 1/6 comes in front first: 5/6 log₃m + 1/2 log₃n. For (iii), the power 3/2 comes in front: 3/2 (log 2 + log x).
Question 9 — 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), ∛68.24, log x = 1/3 × 1.8340 = 0.6113, and antilog gives x = 4.086. For (ii), 319.8 × 3.543, times becomes plus, so log x = 2.5049 + 0.5494 = 3.0543, and antilog gives x = 1133.182. For (iii), (36.12 × 750.9) / (113.2 × 9.98), times on top, divide at the bottom, so log x = 1.3804, and antilog gives x = 24.01.
Question 10 — Population of a city. In 2016 the population is 22 million, growing at 2.5% a year, so p(t) = 22(1.025)ᵗ. Set it equal to 35 million: 35 = 22(1.025)ᵗ. Take log of both sides, use the laws, and t = 18.85 — about 19 years. Start year + 19 years: 2016 + 19 = 2035. The population reaches 35 million in the year 2035.
Frequently Asked Questions
What is the difference between Characteristic and Mantissa?
The Characteristic is the whole-number part of a log, and it tells you how big the number is. The Mantissa is the decimal part, and it comes from the log table. Log = Characteristic + Mantissa, and the Mantissa is always positive.
How do I change between log form and power form?
They are the same thing: log_b N = p means b^p = N. Say it like this: “The base jumps up.” The base stays the base, and the power becomes the answer.
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 and 6 to 8. Use the log table when the numbers are ordinary decimals like 68.24 and 319.8 — that is Question 9 and the population sum.
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
Review Exercise 2 pulls the whole chapter together: the MCQs test the ideas, Questions 2 to 5 test the two forms, Questions 6 to 8 test the laws, Question 9 tests the table, and Question 10 shows why it matters in real life. Download the free PDF, check your answers in the table at the end, and Chapter 2 is finished. For the laws that power this review, see Exercise 2.4, 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 • 14 Pages • Class 9 Maths • Review Exercise 2