Class 9 Maths Chapter 2 Logarithms Notes – Exercise 2.1 Solved (Free PDF Download)

Chapter 2 of Class 9 Mathematics is Logarithms, and Exercise 2.1 is its gentle doorway: scientific notation. Before any logarithm appears, the chapter teaches a shortcut for writing enormous and tiny numbers — the speed of light, the size of planets — in the compact form a × 10ⁿ. It follows the completion of Chapter 1, whose four posts end with the Chapter 1 Review Exercise.

These notes cover Exercise 2.1 completely, in handwritten style and in very simple words. The whole exercise is taught as one game — hopping the decimal point and counting the hops — then played in both directions across all six questions, with a check built into nearly every answer.

Read This First: One Game, Played Both Directions

Let us be honest with each other. Scientific notation frightens students out of all proportion to its difficulty, usually because it is taught as two lists of rules — one for big numbers, one for small numbers, plus special cases. It is not two lists. It is one game. The decimal point hops until exactly one digit stands before it; the number of hops becomes the power of 10; hops left make the power positive, hops right make it negative. Question 1 plays the game forward, Question 2 plays it backwards, and Questions 3 to 6 dress the same hops in real-life costumes — light, Earth and Mars.

These notes never ask you to memorize which case you are in. Every solution starts by asking where the point must land, then counts hops out loud. Two fix-up parts in Question 1 — where the number already wears a ×10 costume in the wrong size — are handled by the same hop-counting, adjusting the power instead of restarting. And because this book uses the words “standard form” to mean the full ordinary number, a full warning box is devoted to that naming trap, the only place this easy exercise can actually cost marks.

Notes Details

Class	9
Subject	Mathematics
Chapter	2 – Logarithms
Exercise	2.1 (complete)
Format	PDF (handwritten notes)
Pages	7
Language	Simple English
Price	Free

What Is Included in These Notes?

1. Into Scientific Notation (Question 1)

All six parts: two big numbers (2000000 → 2 × 10⁶ and 48900 → 4.89 × 10⁴), two tiny numbers (0.0042 → 4.2 × 10⁻³ and 0.0000009 → 9 × 10⁻⁷), and the two fix-ups where the costume is the wrong size (73 × 10³ → 7.3 × 10⁴ and 0.65 × 10² → 6.5 × 10¹). Each answer is checked by converting it back.

2. Back to Ordinary Notation (Question 2)

The game in reverse: 8.04 × 10² = 804, 3 × 10⁵ = 300000, 1.5 × 10⁻² = 0.015, 1.77 × 10⁷ = 17700000, 5.5 × 10⁻⁶ = 0.0000055 and 4 × 10⁻⁵ = 0.00004 — with a place-counting self-check for the tiny ones, where zeros are easiest to lose.

3. The Two Names Trap

A warning box on vocabulary: in this book’s questions, “scientific notation” means the a × 10ⁿ form, while “standard form” (Questions 3, 5 and 6) means the full ordinary number. Plus the zero-counting shortcut — the power of 10 is exactly its number of zeros.

4. Real-Life Giants (Questions 3 to 6)

The speed of light written out as 300000000 m/s, Earth’s circumference dressed up as 4.0075 × 10⁷ m (with a note on why its middle zeros must be kept), and the diameters of Mars (6779 km) and Earth (12756 km) — finished with a sense-check comparing the two planets. The notes close with a one-page revision sheet and four try-yourself questions with answers.

Exercise 2.1 Detailed Walkthrough: Every Question Explained

Question 1 — Into scientific notation. Every part is the same two decisions: where must the point land, and how many hops does it take to get there? For 2000000 the point starts at the very end and hops left six times to land after the 2, giving 2 × 10⁶ — notice that the answer keeps only the meaningful digit and lets the power carry all six zeros. For 48900 the landing place is after the 4, four hops away, and the digits that follow the point are kept while the trailing zeros fall away: 4.89 × 10⁴. The tiny numbers reverse everything: the point hops right, and each rightward hop drives the power below zero. So 0.0042 becomes 4.2 × 10⁻³ after three hops, and 0.0000009 — a number with six zeros guarding a single 9 — becomes 9 × 10⁻⁷ after seven hops, which is why the notes teach the shortcut of counting the guarding zeros and adding one. Parts (v) and (vi) are the fix-ups, where the costume is already on but the wrong size. In 73 × 10³ the first number has two digits before the point, one too many; shrinking 73 into 7.3 costs one leftward hop, and the power must grow by one to keep the value honest: 7.3 × 10⁴. In 0.65 × 10² the first number is below 1; growing it to 6.5 spends one rightward hop, and the power drops by one: 6.5 × 10¹. Both fix-ups are proved by value — 73 × 10³ and 7.3 × 10⁴ are both 73000, and 0.65 × 10² and 6.5 × 10¹ are both 65.

Question 2 — Back to ordinary notation. The reverse journey obeys the sign. A positive power means the number was big, so the point hops right and the number grows, swallowing zeros as needed: 8.04 × 10² = 804, where the hop simply walks past the point; 3 × 10⁵ = 300000, where five hops must be filled with five zeros because the 3 has no digits of its own to walk through; and 1.77 × 10⁷ = 17700000, a mixture — two hops through its own digits, five more through fresh zeros. A negative power means the number was tiny, so the point hops left and the number shrinks into the decimal places: 1.5 × 10⁻² = 0.015, 4 × 10⁻⁵ = 0.00004, and the most error-prone of all, 5.5 × 10⁻⁶ = 0.0000055. For these the notes teach a counting check that never fails: the exponent’s size is the decimal place where the first real digit must land — sixth place for 10⁻⁶ — so a student who writes five zeros before the 5 can count the places and catch the slip immediately.

Question 3 — The speed of light. Here “standard form” means the number written out in full, and 3 × 10⁸ unwraps by giving the 3 its eight zeros: 300000000 metres per second. The walkthrough reads the answer in words — three hundred million metres in a single second — because attaching a spoken size to a string of zeros is how students build a feel for whether an answer is even plausible.

Question 4 — Around the Earth. The circumference of the Earth at the equator, 40075000 m, travels the other way: seven leftward hops put one digit in front, and the answer is 4.0075 × 10⁷ m. This question carries the subtlest point of the exercise — which digits survive the journey. The zeros inside 40075 are not decoration; they hold the places between the 4 and the 75, so they must appear inside the answer as 4.0075. A student who “tidies” them away writes 4.75 × 10⁷ and has quietly enlarged the Earth by seven million metres.

Questions 5 and 6 — Mars and Earth. Two standard-form unwraps: the diameter of Mars, 6.779 × 10³ km, becomes 6779 km after three rightward hops, and the diameter of Earth, 1.2756 × 10⁴ km, becomes 12756 km after four. The walkthrough closes by putting the two answers side by side and testing them against reality — double Mars would be 13558 km, slightly more than Earth’s 12756, so Earth is almost twice as wide as Mars, which is true. Paired questions deserve paired checks, and a swapped power anywhere would have broken this comparison loudly.

Why These Notes Help Slow Learners

This exercise is short, but it is where Chapter 2 either becomes friendly or becomes frightening, because every later logarithm question assumes this notation feels automatic. These notes make it automatic by refusing to split it into cases: one hopping game, one counting habit, both directions. Checks are woven into the fabric — fix-ups are verified by value, tiny answers by place-counting, planets by comparison — so the student practises self-correction instead of answer-copying. And the two genuine mark-losers of the exercise, the standard-form naming trap and the dropped middle zero, are flagged in their own warning boxes rather than buried in working.

How to Use These Notes

Read page one until you can say the shape from memory: one digit before the point, and 10ⁿ carries the zeros. Then work through Questions 1 and 2 in pairs — convert a number, then convert your answer back; if you do not land where you started, recount the hops before looking at the solution. Learn the four real-life giants by heart (light, Earth’s girth, Earth’s width, Mars) as anchors. The night before a test, the one-page revision sheet and the four try-yourself questions are enough.

Download Class 9 Maths Logarithms Notes PDF

The complete Exercise 2.1 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 2.1

1. Leaving two digits before the point. 48.9 × 10³ is not scientific notation — the first number must sit between 1 and 10. Hop once more: 4.89 × 10⁴.

2. Giving tiny numbers a positive power. 0.0042 is smaller than 1, so its power must be negative: 4.2 × 10⁻³. If your power’s sign disagrees with your number’s size, the sign is wrong.

3. Dropping the middle zeros. Earth’s circumference is 4.0075 × 10⁷, not 4.75 × 10⁷. Zeros between real digits hold places — they are part of the number.

4. Swapping “standard form” and “scientific notation”. In this book, Questions 3, 5 and 6 use “standard form” for the full ordinary number. Read what the question starts from before deciding which direction to travel.

5. Losing a zero on the way back. 5.5 × 10⁻⁶ = 0.0000055, with the 5 in the sixth decimal place. Count the places against the power every time you write a shrinking answer.

Frequently Asked Questions

Are these Class 9 Maths notes free?

Yes. The Exercise 2.1 PDF is completely free to download, print and share. Everything on All Free Assets is free.

Which exercise is covered in these notes?

Exercise 2.1 of Chapter 2 (Logarithms) from the Class 9 Mathematics textbook — all six questions: expressing numbers in scientific notation, expressing them back in ordinary notation, and the four real-life questions about light, Earth and Mars.

Why does a Logarithms chapter begin with scientific notation?

Because logarithms are built on powers of 10. Scientific notation makes those powers visible — the exponent in 3 × 10⁸ is exactly the kind of number a logarithm will soon ask for. Mastering Exercise 2.1 makes the rest of Chapter 2 far easier.

Are these notes suitable for slow learners?

Yes, they are written especially for slow learners: the whole exercise is taught as one decimal-point hopping game, every answer carries a check, the naming 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 2.1 is the friendliest possible start to Logarithms: one game of hopping the decimal point, played forward into scientific notation and backward into ordinary numbers, dressed in the real sizes of planets and the speed of light. Download the free PDF, practise until the hop-counting is automatic, and the logarithms that follow will meet a student who already thinks in powers of ten. For the completed Chapter 1, see the Real Numbers Review Exercise, and find everything on our Free Class 9 Notes and Free Math Notes pages — everything on AllFreeAssets.com is free, always.