Introduction
HSA Notes brings short, clear, exam focused notes designed to help Class 9 SSC Part One students score 90 plus marks in Punjab Board exams. This page covers Unit 2, Logarithms, spanning Exercise 2.1 through Exercise 2.4, along with the Review Exercise, giving a complete, fast revision resource for the entire unit.
🎓 Online Classes with Experienced Experts
HSA Notes Features
- Very short, easy to remember definitions.
- Step by step solved answers for every question.
- Quick tricks and memory aids for faster revision.
- Exact alignment with the PCTB and PECTAA textbook.
- Scientific notation and log conversions simplified into clear methods.
- Past paper pattern questions included throughout.
- A fast revision format built for last minute review.
- Trusted by thousands of Punjab students.
Available Across All Punjab Boards and Top Schools
These notes are built for students across all Punjab Boards, including Lahore, Gujranwala, Rawalpindi, Faisalabad, Bahawalpur, Multan, Sargodha, Sahiwal, and DG Khan, and are used at top schools including The City School, Beaconhouse School System, Roots International Schools, Lahore Grammar School, Karachi Grammar School, Froebel’s International School, The Educators, Army Public School, Divisional Public School, and Government Model School.
Official Youtube Channel : www.youtube.com/@PECTAAlearnersclub
Exercise 2.1, Scientific Notation
Express in scientific notation
| No. | Number | Scientific Notation |
|---|---|---|
| i | 2,000,000 | 2.0 times 10 to the power 6 |
| ii | 48,900 | 4.89 times 10 to the power 4 |
| iii | 0.0042 | 4.2 times 10 to the power negative 3 |
| iv | 0.0000009 | 9.0 times 10 to the power negative 7 |
| v | 73 times 10 cubed | 7.3 times 10 to the power 4 |
| vi | 0.65 times 10 squared | 6.5 times 10 to the power 1 |
Express in ordinary notation
| No. | Scientific Notation | Ordinary Notation |
|---|---|---|
| i | 8.04 times 10 squared | 804 |
| ii | 3 times 10 to the power 5 | 300,000 |
| iii | 1.5 times 10 to the power negative 2 | 0.015 |
| iv | 1.77 times 10 to the power 7 | 17,700,000 |
| v | 5.5 times 10 to the power negative 6 | 0.0000055 |
| vi | 4 times 10 to the power negative 5 | 0.00004 |
Applied examples
- Speed of light, 3 times 10 to the power 8 meters per second, equals 300,000,000 meters per second.
- Earth’s circumference, 40,075,000 meters, equals 4.0075 times 10 to the power 7 meters.
- Mars diameter, 6.779 times 10 cubed kilometers, equals 6,779 kilometers.
- Earth diameter, 1.2756 times 10 to the power 4 kilometers, equals 12,756 kilometers.
Exercise 2.2, Logarithmic and Exponential Form
Express in logarithmic form
| No. | Exponential Statement | Logarithmic Form |
|---|---|---|
| i | 10 cubed equals 1000 | log base 10 of 1000 equals 3 |
| ii | 2 to the 8th equals 256 | log base 2 of 256 equals 8 |
| iii | 3 to the negative 3 equals 1 over 27 | log base 3 of 1 over 27 equals negative 3 |
| iv | 20 squared equals 400 | log base 20 of 400 equals 2 |
| v | 16 to the negative one quarter equals one half | log base 16 of one half equals negative one quarter |
| vi | 11 squared equals 121 | log base 11 of 121 equals 2 |
| vii | p equals q to the power r | log base q of p equals r |
| viii | 32 to the negative one fifth equals one half | log base 32 of one half equals negative one fifth |
Express in exponential form
| No. | Logarithmic Statement | Exponential Form |
|---|---|---|
| i | log base 5 of 125 equals 3 | 5 cubed equals 125 |
| ii | log base 2 of 16 equals 4 | 2 to the 4th equals 16 |
| iii | log base 23 of 1 equals 0 | 23 to the 0 equals 1 |
| iv | log base 5 of 5 equals 1 | 5 to the 1st equals 5 |
| v | log base 2 of one eighth equals negative 3 | 2 to the negative 3 equals one eighth |
| vi | log base 9 of 3 equals one half | 9 to the one half equals 3 |
| vii | log of 100,000 equals 5 | 10 to the 5th equals 100,000 |
| viii | log base 4 of one sixteenth equals negative 2 | 4 to the negative 2 equals one sixteenth |
Exercise 2.3, Characteristics, Mantissa, and Antilogs
Find the characteristic
| No. | Number | Characteristic |
|---|---|---|
| i | 5,287 | 3 |
| ii | 59.28 | 1 |
| iii | 0.0567 | negative 2 |
| iv | 234.7 | 2 |
| v | 0.000049 | negative 5 |
| vi | 145,000 | 5 |
Find the logarithm
| No. | Value | Approximate Log |
|---|---|---|
| i | 43 | 1.6335 |
| ii | 579 | 2.7627 |
| iii | 1.982 | 0.2971 |
| iv | 0.0876 | negative 1.0575 |
| v | 0.047 | negative 1.3279 |
| vi | 0.000354 | negative 3.4510 |
From a given logarithm
| No. | Statement |
|---|---|
| i | log 3177 equals 3.5019 |
| ii | log 31.77 equals 1.5019 |
| iii | log 0.03177 equals negative 1.4981 |
Find x, antilog
| No. | Given | x |
|---|---|---|
| i | log x equals 0.0065 | approximately 1.015 |
| ii | log x equals 1.192 | approximately 15.55 |
| iii | log x equals negative 3.434 | approximately 0.000368 |
| iv | log x equals negative 1.5726 | approximately 0.02676 |
| v | log x equals 4.3561 | approximately 22,700 |
| vi | log x equals negative 2.0184 | approximately 0.00959 |
Exercise 2.4, Laws of Logarithms
Evaluate
| No. | Expression | Result |
|---|---|---|
| i | log base 2 of 18 minus log base 2 of 9 | 1 |
| ii | log base 2 of 64 plus log base 2 of 2 | 7 |
| iii | one third log base 3 of 8 minus log base 3 of 18 | negative 2 |
| iv | 2 log 2 plus log 25 | 2 |
| v | one third log base 4 of 64 plus 2 log base 5 of 25 | 5 |
| vi | log base 3 of 12 plus log base 3 of 0.25 | 1 |
Write as a single logarithm
| No. | Expression | Single Log Form |
|---|---|---|
| i | one half log 25 plus 2 log 3 | log 45 |
| ii | log 9 minus log one third | log 27 |
| iii | log base 5 of b squared, times log base a of 5 cubed | 6 log base a of b |
| iv | 2 log base 3 of x plus log base 3 of y | log base 3 of x squared y |
| v | 4 log base 5 of x minus log base 5 of y plus log base 5 of z | log base 5 of x to the 4th, z, over y |
| vi | 2 ln a plus 3 ln b minus 4 ln c | ln of a squared b cubed over c to the 4th |
Expand
| No. | Expression | Expanded Form |
|---|---|---|
| i | log of 11 over 5 | log 11 minus log 5 |
| ii | log base 5 of the quantity 8a to the 6th, all to the one half | three halves log base 5 of 2, plus 3 log base 5 of a |
| iii | ln of a squared b over c | 2 ln a plus ln b minus ln c |
| iv | log of xy over z, all to the one ninth | one ninth times the quantity log x plus log y minus log z |
| v | ln of 16x cubed, all to the one third | four thirds ln 2, plus ln x |
| vi | log base 2 of the quantity 1 minus a over b, all to the 5th | 5 times the quantity log base 2 of 1 minus a, minus log base 2 of b |
Find x
| No. | Equation | x |
|---|---|---|
| i | log 2 plus log x equals 1 | 5 |
| ii | log base 2 of x plus log base 2 of 8 equals 5 | 4 |
| iii | 81 to the x equals 243 to the x plus 2 | negative 10 |
| iv | one twenty seventh to the x minus 6, equals 27 | 5 |
| v | log of 5x minus 10 equals 2 | 22 |
| vi | log base 2 of x plus 1, minus log base 2 of x minus 4, equals 2 | 17 over 3 |
Values using a log table
| No. | Expression | Approximate Value |
|---|---|---|
| i | 3.68 times 4.21, over 5.234 | approximately 2.954 |
| ii | 4.67 times 2.11 times 2.397 | approximately 23.458 |
| iii | 20.46 squared, times 2.4122, over 754.3 | approximately 1.327 |
| iv | the cube root of 9.364 times 21.64 over 3.21 | given as approximately 14.138 |
A note on the last entry above: solving this expression directly gives 9.364 times 21.64, divided by 3.21, which comes to approximately 63.1, and the cube root of 63.1 is approximately 3.98, not 14.138. This looks like a possible transcription issue in the original source rather than an error in the underlying method, so it is worth checking this specific value against your textbook or the original PDF before relying on it.
Applied word problems
- Earthquake magnitude works out to 3.
- An investment doubles in approximately 14.2 years.
- Temperature at 500 meters works out to approximately 17.18 degrees Celsius.
Review Exercise 2
Multiple Choice Questions
| No. | Answer |
|---|---|
| i | c, 5,200,000 |
| ii | b, 3.4 times 10 to the power negative 4 |
| iii | b, 10 |
| iv | d, 3 |
| v | a, 2 |
| vi | c, 2.3010 |
| vii | d, undefined |
| viii | c, 4 |
| ix | d, log 15 |
| x | c, log base 3 of 81 equals 4 |
Additional Review Answers
- Scientific notation: (i) 5.67 times 10 to the power negative 4, (ii) 7.34 times 10 squared.
- Ordinary notation: (i) 2,600, (ii) 0.0008794, (iii) 0.000006.
- Logarithmic form: (i) log base 3 of 2187 equals 7, (ii) log base a of c equals b, (iii) log base 12 of 144 equals 2.
- Exponential form: (i) 4 to the x equals 8, (ii) 9 cubed equals 729, (iii) 4 to the 5th equals 1024.
- Find x: (i) x equals 3, (ii) x equals negative one half, (iii) x equals negative 0.6.
- Single logarithm: (i) log of x to the 7th over y to the 6th, (ii) log 2, (iii) log base 5 of 2.
- Expand: (i) log x plus log y plus 6 log z, (ii) five sixths log base 3 of m, plus one half log base 3 of n, (iii) three halves log 2, plus three halves log x.
- Values: (i) approximately 4.085, (ii) approximately 1,133, (iii) approximately 24.02.
- Population works out to approximately 2,035.
View the full PDF for diagrams and constructions supporting every answer above.
Why This Unit Matters for Your Overall Score
Logarithms is a unit that rewards careful, methodical work far more than memorization, since almost every question relies on a small set of core rules, the product rule, the quotient rule, and the power rule, applied correctly and in the right order.
Getting comfortable converting between logarithmic and exponential form early in Exercise 2.2 makes every later exercise easier, since solving for x in Exercise 2.4 almost always comes back to that same conversion. Since this unit also connects directly to scientific notation, a skill used throughout science subjects as well, time spent mastering Unit 2 pays off well beyond the Mathematics paper alone.
Join Online Classes
Join online classes to aim for 90 plus marks, including past paper practice and doubt clearing sessions.
Contact: hsanotes48@gmail.com
Frequently Asked Questions
Which exercises are covered in these Unit 2 notes? Exercise 2.1, Exercise 2.2, Exercise 2.3, Exercise 2.4, and the Review Exercise are all covered in full.
Does this unit cover both logarithms and scientific notation? Yes, Exercise 2.1 covers scientific notation, while Exercise 2.2 through 2.4 and the Review Exercise cover logarithmic form, characteristics and mantissa, antilogs, and the laws of logarithms.
Are log table based calculations included? Yes, characteristic and mantissa calculations, along with antilog and log table based value problems, are covered in Exercise 2.3 and 2.4.
Are these notes aligned with the PCTB and PECTAA textbook? Yes, these notes are built to exactly match the PCTB and PECTAA New Book for Class 9 Mathematics.
Conclusion
With Exercise 2.1 through 2.4 and the Review Exercise fully solved, from scientific notation through the laws of logarithms, Class 9 students have a complete, exam ready resource for Unit 2. Use these notes alongside regular practice with past papers to build the confidence needed for 90 plus marks in your Punjab Board exam.
Best of luck for your Class 9 Mathematics exam!