Exercise 1.4 Class 12 Math Solutions - Unit 1 Graphical Representation of Functions, Punjab Board
Step-by-step solutions to Exercise 1.4, Unit 1: Graphical Representation of Functions, for Class 12 Punjab Board students. Covers solving exponential equations, logarithmic equations, exponential inequalities, and logarithmic inequalities with domain restrictions and graphical validation, worked out question by question.
Exercise Overview
Exercise 1.4 of Class 12 Mathematics focuses on the algebraic and graphical solutions of exponential and logarithmic equations and inequalities under the Punjab Board PCTB / PECTAA curriculum. Students learn systematic algebraic methods to isolate exponential terms, apply natural or common logarithms to both sides, and solve equations with differing bases (such as a^(f(x)) = b^(g(x))) using logarithm properties.
In addition to standard equations, a substantial portion of this exercise addresses logarithmic and exponential inequalities (such as logₐ(x) > k or e^(2x) ≤ c). Solving logarithmic inequalities requires careful attention to base conditions: when the base a > 1, the inequality direction is preserved, whereas for 0 < a < 1, the inequality reverses. Crucially, students must always verify the strict positivity domain constraint of logarithmic arguments (x > 0) to eliminate extraneous solutions that do not satisfy the original functional domain.
For Punjab Board HSSC Part 2 examinations across all nine educational boards (Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, Bahawalpur, Sargodha, Sahiwal, and D.G. Khan), Exercise 1.4 is a frequent source of medium and long exam problems, along with tricky MCQ items on inequality intervals. These step-by-step solutions provide rigorous working, domain verification, and clear algebraic justifications for each problem.