Class 9 Maths Chapter 2 Logarithms Notes – Exercise 2.3 Solved (Free PDF Download)

Exercise 2.3 is where Logarithms stops being a new word and starts being a tool you can actually use. Exercise 2.1 taught scientific notation, and Exercise 2.2 taught the flip between power form and log form. Now, in Exercise 2.3, you learn to find the logarithm of any number — and to go backwards and find the number again. The whole exercise runs on two new words, Characteristic and Mantissa, and once you know what each one does, every question in this exercise is the same few moves repeated. These notes cover Exercise 2.3 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: One Log Has Two Parts

A logarithm answer is never one number. It is always two parts added together, and each part has a real name. The first part is the Characteristic. It is a whole number — like 3, or 0, or −2 — and it tells you how big the number is. The second part is the Mantissa. It is a small decimal — like 0.5019 — and it comes from the log table book. The rule for the whole exercise is one short line: log = Characteristic + Mantissa. Say it out loud twice: “Log is Characteristic plus Mantissa.” That is the whole idea, and everything else in this exercise is just finding those two parts. The Characteristic you can find yourself, with a quick trick. The Mantissa you read from the table — you never have to learn it by heart. And one rule you must never break: the Mantissa is always plus. Never minus. When a logarithm looks negative, you fix it first, and Question 4 shows you exactly how.

Notes Details

Class	9 / Grade 9
Subject	Mathematics
Chapter	2 – Logarithms
Exercise	2.3 (complete)
Format	PDF (handwritten notes)
Pages	9
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?

1. The Two Parts of a Log (Pages 1–2)

What Characteristic and Mantissa mean, in plain words: the Characteristic is the whole-number part that tells you how big the number is, and the Mantissa is the decimal part that comes from the log table. The golden rule log = Characteristic + Mantissa, with the worked example log 43 = 1 + 0.6335 = 1.6335.

2. Finding the Characteristic (Page 3)

Two tricks, one for big numbers and one for small numbers. Big number (1 or more): count the digits before the point and take away 1 — so 234.7 has 3 digits before the point, and its Characteristic is 2. Small number (less than 1): count the zeros after the point, add 1, and put a minus — so 0.0567 has 1 zero after the point, and its Characteristic is −2. Big number means a plus Characteristic; small number means a minus Characteristic.

3. Question 1 — Find the Characteristic (Page 4)

All six parts, each done in the same two steps: write the number as a × 10ⁿ, then read n. 5287 gives 3, 59.28 gives 1, 0.0567 gives −2, 234.7 gives 2, 0.000049 gives −5, and 145000 gives 5.

4. Question 2 — Find the Log (Page 5)

All six parts, each done in the same three steps: find the Characteristic, get the Mantissa from the table, and add them. log 43 = 1.6335, log 579 = 2.7627, log 1.982 = 0.2971, log 0.0876 = −1.0575, log 0.047 = −1.3279, and log 0.000354 = −3.4510.

5. Question 3 — Same Digits, Same Mantissa (Page 6)

Given log 3.177 = 0.5019, find log 3177, log 31.77 and log 0.03177. The digits never change — 3, 1, 7, 7 — so the Mantissa never changes either. Only the point moves, so only the Characteristic changes: 3.5019, 1.5019 and −1.4981.

6. Question 4 — Find x with Antilog (Pages 7–8)

The opposite of log: you are given log x and you find x. The hard part is fixed first — when log x is minus, like −3.434, you rewrite it as −4 + 0.566, so the Mantissa is plus again. Then three steps: split into Characteristic and Mantissa, take the antilog of the Mantissa, and place the point. All six parts: x = 1.015, 15.56, 0.0003681, 0.02675, 22704 and 0.009585.

7. Revision Sheet (Page 9)

The whole exercise in five lines, the two mistakes students make most, a try-it-yourself box, and the full check-your-answers table.

Exercise 2.3 Detailed Walkthrough: Every Question Explained

Question 1 — Find the Characteristic. Every part uses the same two steps: write the number as a × 10ⁿ, then read n — n is the answer. For 5287, that is 5.287 × 10³, so the Characteristic is 3. For 59.28, it is 5.928 × 10¹, so the answer is 1. The small numbers are where students slip, so take them slowly: 0.0567 is 5.67 × 10⁻², so the Characteristic is −2; 0.000049 is 4.9 × 10⁻⁵, so it is −5. And 234.7 is 2.347 × 10², so the answer is 2, while 145000 is 1.45 × 10⁵, so it is 5. The notes end this question with a check you can do in your head every time: is the number big? Then the answer must be plus. Is the number small? Then the answer must be minus.

Question 2 — Find the log. Now the three steps run together: Characteristic, then Mantissa from the table, then add. For 43 the Characteristic is 1 and the Mantissa is 0.6335, so log 43 = 1 + 0.6335 = 1.6335. For 579 the Characteristic is 2 and the Mantissa is 0.7627, so log 579 = 2.7627. For 1.982 the Characteristic is 0 — the number is between 1 and 10, so there is nothing to add — and the Mantissa is 0.2971, so log 1.982 = 0.2971. Then come the small numbers, and the Mantissa rule matters: for 0.0876 the Characteristic is −2 and the Mantissa is 0.9425, so log 0.0876 = −2 + 0.9425 = −1.0575. Notice the Mantissa stayed plus even though the answer is minus — that is the rule working. For 0.047 the Characteristic is −2 and the Mantissa is 0.6721, so log 0.047 = −1.3279. And for 0.000354 the Characteristic is −4 and the Mantissa is 0.5490, so log 0.000354 = −3.4510. The notes remind you not to worry about the Mantissa values: they are given in the table, and you do not need to learn them by heart.

Question 3 — Same digits? Then the Mantissa is the same. You are given log 3.177 = 0.5019, and the digits in 3177, 31.77 and 0.03177 are all the same — 3, 1, 7, 7. So the Mantissa is 0.5019 every time, and only the Characteristic changes. For log 3177 the Characteristic is 3, so the answer is 3 + 0.5019 = 3.5019. For log 31.77 the Characteristic is 1, so the answer is 1 + 0.5019 = 1.5019. For log 0.03177 the Characteristic is −2, so the answer is −2 + 0.5019 = −1.4981. Say it out loud: “Same digits. Same Mantissa.” This is the question examiners love, because it tests whether you understand the idea instead of just reading the table.

Question 4 — Find x. This is antilog: log goes from a number to its log value, and antilog goes back from the log value to the number. The method is three steps: split log x into Characteristic and Mantissa, take the antilog of the Mantissa, and place the point using the Characteristic. The first two parts are gentle: log x = 0.0065 has Characteristic 0 and Mantissa 0.0065, and antilog 0.0065 = 1.015, so x = 1.015. Log x = 1.192 has Characteristic 1 and Mantissa 0.192, and antilog 0.192 gives 15.56, so x = 15.56. Then the hard part arrives: log x = −3.434. The Mantissa must never be minus, so you fix it first: −3.434 = −4 + 0.566. Now the Characteristic is −4 and the Mantissa is +0.566, and antilog 0.566 gives 0.0003681, so x = 0.0003681. The fix is one move: go one step more minus, then write the rest as plus. The same fix works for the rest: log x = −1.5726 becomes −2 + 0.4274, and antilog 0.4274 gives 0.02675. Log x = 4.3561 needs no fix — Characteristic 4, Mantissa 0.3561, antilog 0.3561 gives 22704. And log x = −2.0184 becomes −3 + 0.9816, and antilog 0.9816 gives 0.009585. The quick check at the end is worth learning: x = antilog (log x), always.

Frequently Asked Questions

What is the Characteristic of a logarithm, in simple words?

It is the whole-number part of the log, and it tells you how big the number is. It is also the power of 10 when you write the number as a × 10ⁿ. Big numbers have a plus Characteristic; small numbers have a minus one.

What is the Mantissa?

It is the decimal part of the log, read from the log table book. It depends only on the digits of the number, never on where the point sits — which is why the same digits always give the same Mantissa.

Why must the Mantissa always be positive?

Because the log table only lists positive decimals. If your log value is negative, you rewrite it first — for example, −3.434 becomes −4 + 0.566 — so the Mantissa is plus and the table can be used.

What is an antilog?

It is the opposite of a log. Log takes a number and gives its log value; antilog takes the log value and gives the number back.

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 2.3 gives you the working tools of logarithms: split any log into Characteristic and Mantissa, read the Mantissa from the table, and use antilog to walk back to the number — fixing a negative log first so the Mantissa stays plus. Download the free PDF, practise until the two tricks for the Characteristic are automatic, and the laws of logarithms that follow will feel like the next step, not a new subject. For the flip that came before, see Exercise 2.2 — Power Form and Log Form, and find everything on our Free Class 9 Notes and Free Math Notes pages — everything on AllFreeAssets.com is free, always.

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