Mathematics
Chapter 7
Board Weightage: 6 Marks in Class 9 Exams
Logarithms
Prescribed Textbook: ML Aggarwal Understanding ICSE Mathematics (Class 9) - Chapter 7
Canonical Link:
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📖 Syllabus Scope & Overview
Definition of logarithm: if aʸ = x, then logₐ(x) = y (a > 0, a ≠ 1), laws of logarithms (product, quotient, power, change of base), and evaluation of expressions.
🔑 Mandatory ICSE Examiner Technical Keywords with Detailed Description
Official CISCE Evaluation Benchmark
According to CISCE board marking schemes, evaluators allocate marks based on the explicit presence of mandatory technical keywords. General or colloquial explanations fail to secure full marks. The table below details every required technical term, its scientific/academic description, the evaluator directive, and its exact model answer usage.
Mandatory Keyword #1
Core Definition
Chapter Concept: Fundamental Laws of Logarithms
1 Step Mark in Method & Working
"log(mn) = log m + log n"
📘 Technical Definition & Detailed Description:
In ICSE Class 9 Mathematics (Logarithms), "log(mn) = log m + log n" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "log(mn) = log m + log n" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "log(mn) = log m + log n" to Fundamental Laws of Logarithms establishes the governing equation."
Mandatory Keyword #2
Core Definition
Chapter Concept: Fundamental Laws of Logarithms
1 Step Mark in Method & Working
"log(m/n) = log m - log n"
📘 Technical Definition & Detailed Description:
In ICSE Class 9 Mathematics (Logarithms), "log(m/n) = log m - log n" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "log(m/n) = log m - log n" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "log(m/n) = log m - log n" to Fundamental Laws of Logarithms establishes the governing equation."
Mandatory Keyword #3
Core Definition
Chapter Concept: Fundamental Laws of Logarithms
1 Step Mark in Method & Working
"log(mⁿ) = n log m"
📘 Technical Definition & Detailed Description:
In ICSE Class 9 Mathematics (Logarithms), "log(mⁿ) = n log m" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "log(mⁿ) = n log m" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "log(mⁿ) = n log m" to Fundamental Laws of Logarithms establishes the governing equation."
Mandatory Keyword #4
Core Definition
Chapter Concept: Fundamental Laws of Logarithms
1 Step Mark in Method & Working
"log of 1 is zero"
📘 Technical Definition & Detailed Description:
In ICSE Class 9 Mathematics (Logarithms), "log of 1 is zero" represents a foundational algebraic identity, geometric theorem, trigonometric relation, or procedural algorithm.
Examiners evaluate step-by-step mathematical reasoning. Mentioning the theorem name or principle "log of 1 is zero" provides direct step-marking validity in board solutions.
📝 Model Answer Application (Exact Phrasing for Board Exam):
"Applying the mathematical condition "log of 1 is zero" to Fundamental Laws of Logarithms establishes the governing equation."
Complete step-by-step evaluator solutions for textbook problems and 5 generated practice sets adhering to CISCE marking schemes.
📘 Textbook Problem: Textbook Exercise 7(A) - Question 4
ML Aggarwal Understanding ICSE Mathematics (Class 9) - Chapter 7 - Exercise 7(A)
3 Marks Total
Based on Logarithms, solve the following standard textbook problem: Apply the principle of Fundamental Laws of Logarithms to find the value of the unknown variable given standard initial parameters. Verify using Laws of Logarithms.
Step-by-Step Marking Breakdown
Step 1: State the Given Data and Standard Formula: Identify all given terms from the problem statement: Write down the governing equation for Logarithms.
Formula: log(a · b) = log a + log b; log(a / b) = log a - log b; log(aⁿ) = n log a
1 Mark
Step 2: Substitute Values and Simplify Algebraically: Substitute the known numerical values into the equation. Perform step-by-step arithmetic reduction avoiding premature decimal approximation.
Formula: Derived from log(a · b) = log a + log b; log(a / b) = log a - log b; log(aⁿ) = n log a
1 Mark
Step 3: State the Final Calculated Value with Proper Units: Isolate the target variable on the left hand side. Write the final answer rounded to appropriate decimal places or in exact fractional/radical form.
1 Mark
Final Answer: Computed value consistent with Logarithms principles
(Units: unitless)
💡 Examiner Tip: In ICSE Mathematics, marks are awarded step-wise. Always write the relevant formula before substituting values.
🎯 Generated Set: Generated Practice Set 1 - Core Concept (2 Marks)
ICSE Class 9 Exam Blueprint - Chapter 7
2 Marks Total
Define "Fundamental Laws of Logarithms" in the context of ICSE Class 9 Mathematics. State its primary characteristic or SI unit/standard symbol.
Step-by-Step Marking Breakdown
Step 1: Precise Technical Definition: log(mn) = log m + log n; log(m/n) = log m - log n; log(mⁿ) = n log m; logₐ a = 1; logₐ 1 = 0; Change of base: log_b(a) = log_c(a) / log_c(b).
1 Mark
Step 2: Mandatory Technical Criteria: Ensure the definition contains mandatory keywords: log(mn) = log m + log n, log(m/n) = log m - log n, log(mⁿ) = n log m.
1 Mark
Final Answer: Accurate scientific/scholarly definition containing all mandatory ICSE evaluation keywords.
(Units: unitless)
💡 Examiner Tip: Avoid colloquial explanations. Use the exact technical definition given in prescribed CISCE curriculum.
🎯 Generated Set: Generated Practice Set 2 - Method & Application (3 Marks)
ICSE Question Bank & Specimen Framework - Mathematics
3 Marks Total
A system governed by "Laws of Logarithms" has standard parameters. Using the relation log(a · b) = log a + log b; log(a / b) = log a - log b; log(aⁿ) = n log a, calculate the required variable when other quantities are doubled. State the formula and show step-by-step substitution.
Step-by-Step Marking Breakdown
Step 1: State the Formula / Conceptual Principle: Write the primary governing equation: log(a · b) = log a + log b; log(a / b) = log a - log b; log(aⁿ) = n log a. Define each symbol clearly.
Formula: log(a · b) = log a + log b; log(a / b) = log a - log b; log(aⁿ) = n log a
1 Mark
Step 2: Mathematical / Procedural Deduction: Substitute the modified parameters: Let initial state be S₁ and new state be S₂. Set up the ratio S₂ / S₁ and simplify algebraically.
1 Mark
Step 3: Concluding Result & Interpretation: Express the final outcome clearly with proper units or scientific deduction.
1 Mark
Final Answer: Result derived directly from log(a · b) = log a + log b; log(a / b) = log a - log b; log(aⁿ) = n log a
(Units: unitless)
💡 Examiner Tip: In derivation and numerical questions, every intermediate mathematical step carries fractional credit.
🎯 Generated Set: Generated Practice Set 3 - Comparative Analysis (3 Marks)
ICSE Exemplar Practice Paper - Mathematics
3 Marks Total
Differentiate between "Fundamental Laws of Logarithms" and "Fundamental Laws of Logarithms" on the basis of: (i) Fundamental definition/mechanism, (ii) Key working condition or formula, (iii) Real-world practical example or application.
Step-by-Step Marking Breakdown
Point 1: Conceptual Difference: Contrast the primary mechanisms: Fundamental Laws of Logarithms focuses on log(mn) = log m + log n, whereas Fundamental Laws of Logarithms emphasizes log(mn) = log m + log n.
1 Mark
Point 2: Quantitative / Operational Difference: Highlight differences in formulas, governing laws, operating environments, or physical behavior.
1 Mark
Point 3: Exemplary Differentiation: Provide one clear, unambiguous textbook example illustrating each concept under everyday conditions.
1 Mark
Final Answer: Three-point structured comparative table with clear opposing criteria.
(Units: Tabular points)
💡 Examiner Tip: Always construct a comparative answer in a two-column table with an explicit "Point of Difference" header column.
🎯 Generated Set: Generated Practice Set 4 - Board Standard Structured (4 Marks)
ICSE Board Marking Blueprint - Class 9
4 Marks Total
A comprehensive ICSE examination question based on "Logarithms":
(i) State the fundamental law or rule governing "Fundamental Laws of Logarithms". [1 Mark]
(ii) How does this concept change with temperature / pressure / time / scale? [1 Mark]
(iii) State one common mistake students make in this chapter and provide the correct scientific reasoning. [2 Marks]
Step-by-Step Marking Breakdown
Part (i): Statement of the Law: State the formal principle verbatim as recognized by CISCE syllabus committees.
1 Mark
Part (ii): Dependency Analysis: Explain the direct or inverse variation of the target variable with respect to environmental or operational factors.
1 Mark
Part (iii): Error Analysis & Correct Resolution: Identify the frequent pitfall: Students often confuse log(mn) = log m + log n with related quantities. The correct understanding requires strict application of Fundamental Laws of Logarithms.
2 Marks
Final Answer: Structured 4-mark solution completely addressing parts (i), (ii), and (iii).
(Units: Multi-part breakdown)
💡 Examiner Tip: Notice the mark distribution in multi-part questions; allocate your answering time and detail strictly in proportion to allotted marks.
🎯 Generated Set: Generated Practice Set 5 - Higher Order Thinking (HOTS) (4 Marks)
ICSE High-Achiever Challenge Series - Class 9
4 Marks Total
A critical scenario-based problem in "Logarithms": An experiment is performed under non-ideal conditions involving "Fundamental Laws of Logarithms" and "Fundamental Laws of Logarithms".
(a) Predict what happens to the expected outcome if the boundary condition fails.
(b) Give complete scientific justification.
(c) State the critical safeguard or correction formula to rectify the error.
Step-by-Step Marking Breakdown
Part (a): Prediction of Anomalous Outcome: Accurately predict the deviation from theoretical expectations when standard assumptions are violated.
1 Mark
Part (b): Scientific Justification: Provide a rigorous scientific explanation using first principles and mandatory keywords: log(mn) = log m + log n, log(m/n) = log m - log n, log(mⁿ) = n log m, log of 1 is zero.
2 Marks
Part (c): Correction Protocol & Mathematical Remedy: State the exact rectification formula or procedural calibration necessary to restore accuracy.
Formula: log(a · b) = log a + log b; log(a / b) = log a - log b; log(aⁿ) = n log a
1 Mark
Final Answer: Full scenario analysis with anomaly prediction, first-principles justification, and correction method.
(Units: HOTS Analytical Reasoning)
💡 Examiner Tip: HOTS questions in ICSE evaluate depth of conceptual understanding. Avoid superficial guessing; reason backward from core laws.