Class 9 Maths Chapter 2 Logarithms Notes – Exercise 2.2 Solved (Free PDF Download)

Exercise 2.2 is where Chapter 2 of Class 9 Mathematics — Logarithms — truly begins. Exercise 2.1 warmed up with scientific notation; now the logarithm itself arrives. This exercise teaches the single most important idea of the whole chapter: a power and a logarithm are the same fact written in two costumes, and you can flip freely between them.

These notes cover Exercise 2.2 completely, in handwritten style and in very simple words. Everything is built on one move — the flip — which is taught on page one, practised in both directions through Questions 1 and 2, and then used to hunt down the unknown x in Question 3.

Read This First: One Flip Runs the Whole Exercise

A logarithm is only a question: “What power must I put on this base to get that number?” Because 2 × 2 × 2 = 8, the power is 3 — and that 3 is the logarithm. Written formally, a^b = c means exactly the same as log_a c = b. Nothing is added and nothing is lost; it is one fact in two costumes.

The flip has three simple moves. The base never moves house: the base of the power becomes the little base of the log. The result number sits after the word log. And the power jumps over to become the answer — which is why the answer to every logarithm question is always a power, never a big scary number. Question 1 practises the flip from power form into log form, Question 2 practises it backwards, and Question 3 hides an x in one of the three seats — base, answer or number — and asks you to flip first and catch it second. Students who learn this one move properly find the rest of Chapter 2 surprisingly friendly.

Notes Details

What Is Included in These Notes?

1. The Flip Rule (Page 1)

What a logarithm actually is, in plain words: a question asking for a power. The one rule of the exercise — a^b = c is the same fact as log_a c = b — with the three moves named (base stays base, result goes after log, power becomes the answer) and the question trick for reading any log out loud.

2. Powers Become Logs (Question 1)

All eight parts flipped into logarithmic form: 10³ = 1000, 2⁸ = 256, 20² = 400 and 11² = 121; the negative powers 3⁻³ = 1/27, 16^−1/4 = 1/2 and 32^−1/5 = 1/2; and the letter part p = q^r, which flips exactly like numbers. A five-second self-check — base to the power of the answer should return the result — is taught with the very first part.

3. Logs Become Powers (Question 2)

All eight parts flipped back into exponential form, including the two golden facts every examiner loves — the log of 1 is always 0 (log₂₃ 1 = 0) and the log of the base itself is always 1 (log₅ 5 = 1) — and the back-to-front trap: parts like 1/2 = log₉ 3 and 5 = log₁₀ 100000, where the sentence must be turned around before flipping.

4. Find x (Question 3) + Revision

All six parts, organised by the three hiding places of x: as the base (log_x 64 = 3, log_x 8 = 1), as the answer (log₅ 1 = x, log₂ 1024 = x — solved with a halving ladder), and as the number (log₁₀ x = −3, log₄ x = 3/2). The notes close with a one-page revision sheet and four try-yourself questions with answers.

Exercise 2.2 Detailed Walkthrough: Every Question Explained

Question 1 — Express in logarithmic form. Every part is the same three moves: keep the base, place the result after log, and let the power become the answer. For 10³ = 1000 the pieces are base 10, power 3, result 1000, and they land as log₁₀ 1000 = 3. The same choreography gives log₂ 256 = 8 from 2⁸ = 256, log₂₀ 400 = 2 from 20² = 400 — proof that even an unusual base like 20 changes nothing — and log₁₁ 121 = 2 from 11² = 121. The parts students fear are the negative powers, but the flip treats them identically: 3⁻³ = 1/27 becomes log₃ 1/27 = −3, the minus sign travelling with the power into the answer, exactly as it should, because a negative power is precisely what shrinks a result into a fraction. The fraction powers behave the same way: 16^−1/4 = 1/2 becomes log₁₆ 1/2 = −1/4, and 32^−1/5 = 1/2 becomes log₃₂ 1/2 = −1/5 — the whole power, copied exactly, becomes the answer. The notes pause on that last one to show why it is true: 32 is five 2s multiplied together, so the fifth root of 32 is 2, and the negative power flips it to 1/2. Part (vii) replaces numbers with letters — p = q^r — and the walkthrough stresses that letters are just numbers in disguise: the base is whichever quantity carries the power, here q, so the answer is log_q p = r.

Question 2 — Express in exponential form. Now the flip runs backwards, and the pieces trade places in reverse: the log’s answer becomes the power. So log₅ 125 = 3 becomes 5³ = 125, and log₂ 16 = 4 becomes 2⁴ = 16. Two parts are gifts in disguise. In log₂₃ 1 = 0 the answer is 23⁰ = 1, and the notes elevate it into a golden fact: any base to the power 0 gives 1, so the logarithm of 1 is 0 for every base that will ever exist. In log₅ 5 = 1 the answer is 5¹ = 5, the second golden fact: the log of the base itself is always 1, because a number to the power 1 is itself. The negative answers continue the pattern of Question 1 in reverse — log₂ 1/8 = −3 becomes 2⁻³ = 1/8, and log₄ 1/16 = −2 becomes 4⁻² = 1/16, each checkable in seconds (4² = 16, so 4⁻² must be 1/16). The two back-to-front parts get their own warning: in 1/2 = log₉ 3 and 5 = log₁₀ 100000 the log sits on the right-hand side, and the safe routine is to turn the sentence around first — an equals sign works both ways — and only then flip, giving 9^1/2 = 3 and 10⁵ = 100000. The first of these also quietly teaches that the power 1/2 means square root, and indeed √9 = 3.

Question 3 — Find the value of x. The unknown hides in one of three seats, and the notes sort the six parts by seat rather than by order, because each seat has its own simple capture method. When x is the base, flip and compare powers: log_x 64 = 3 becomes x³ = 64, and splitting 64 by repeated division (64 ÷ 4 = 16, 16 ÷ 4 = 4, 4 ÷ 4 = 1) shows 64 = 4³, so x = 4; log_x 8 = 1 becomes x¹ = 8, and since power 1 changes nothing, x = 8 immediately. When x is the answer, write the number as a power of the base: log₅ 1 = x becomes 5^x = 1 = 5⁰, so x = 0 — the golden fact returning as a free mark — while log₂ 1024 = x becomes 2^x = 1024, solved by the halving ladder: 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1 is ten jumps, so x = 10. When x is the number, the flip hands it over directly, because the power side of the flipped equation is x itself: log₁₀ x = −3 gives x = 10⁻³ = 1/1000, and log₄ x = 3/2 gives x = 4^3/2, the one part needing a bridge — write 4 as 2², stack the powers and multiply them, 2 × 3/2 = 3, so x = 2³ = 8.

Why These Notes Help Slow Learners

Logarithms frighten students mainly because they arrive as a new notation before they arrive as an idea. These notes reverse that order: the idea — a log is a question asking for a power — comes first, on page one, in words a ten-year-old could follow, and the notation is then introduced as nothing more than the written form of that question. One flip replaces the two memorised lists (one for “to log form”, one for “to exponential form”) that textbooks usually demand, and every question type is sorted by a visible feature — where is x hiding? is the sentence back-to-front? — so the student always knows which small move comes next. Checks are built into the habit from the first page: raise the base to your answer and see whether the result comes back.

How to Use These Notes

Read page one until you can say the flip from memory and explain it in your own words: base stays base, power becomes the answer. Then work Questions 1 and 2 as pairs — flip a power into a log, then flip your answer back; if you do not land where you started, find the piece that moved wrongly before peeking at the solution. Learn the two golden facts by heart (log of 1 is 0; log of the base itself is 1), practise the halving ladder on 1024 until it takes seconds, and finish with the revision page and the four try-yourself questions the night before a test.

Download Class 9 Maths Logarithms Notes PDF

The complete Exercise 2.2 notes are available as a free PDF — the same handwritten pages described above, ready to read on your phone, print for your notebook, or share with your class fellows. No sign-up, no payment, just the download.

5 Common Mistakes Students Make in Exercise 2.2

1. Moving the base. For 2⁸ = 256, students sometimes write log₈ 256 = 2. The base never moves house: it is log₂ 256 = 8. The base of the power is the base of the log, always.

2. Dropping the negative sign. 3⁻³ = 1/27 flips to log₃ 1/27 = −3, not 3. The power becomes the answer exactly as it is — minus sign included.

3. Panicking at fraction powers. Powers like −1/4 and −1/5 look frightening but need no calculation in Question 1: they are simply copied, whole, into the answer’s seat.

4. Flipping a back-to-front sentence without turning it around. In 1/2 = log₉ 3, first rewrite it as log₉ 3 = 1/2, then flip to 9^1/2 = 3. Flipping the unturned sentence scrambles the base and the answer.

5. Trying to solve for x before flipping. In log_x 64 = 3, guessing at the log form produces nonsense like x = 64 ÷ 3. Always flip into power form first — x³ = 64 — where the unknown can actually be caught.

Frequently Asked Questions

Are these Class 9 Maths notes free?

Yes. The Exercise 2.2 PDF is completely free to download, print and share. Everything on All Free Assets is free.

Which exercise is covered in these notes?

Exercise 2.2 of Chapter 2 (Logarithms) from the Class 9 Mathematics textbook — all three questions: expressing powers in logarithmic form, expressing logarithms in exponential form, and finding the value of x in logarithmic equations.

What is a logarithm, in simple words?

A logarithm is a question: “What power must I put on this base to get that number?” If 2³ = 8, then log₂ 8 = 3. The answer to a log is always a power.

Are these notes suitable for slow learners?

Yes, they are written especially for slow learners: the whole exercise is taught as one flip rule, every solution names its pieces before moving them, the back-to-front trap gets its own warning box, and the revision sheet fits on one page.

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.

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 2.2 is the heart of Chapter 2: one flip between power form and log form, practised in both directions, and then used to catch an unknown hiding in three different seats. Download the free PDF, practise until the flip is automatic, and the laws of logarithms that follow will feel like variations on a game you already know. For the warm-up that came before, see Exercise 2.1 — Scientific Notation, and find everything on our Free Class 9 Notes and Free Math Notes pages — everything on AllFreeAssets.com is free, always.