This article covers the essential skills of logarithmic functions and their graphs – an advanced topic in DSE Paper 1 Section A(2) and Section B worth 4–7 marks. You will learn the definition of logarithms as the inverse of exponentials, their properties (domain, range, asymptote, monotonicity), how to sketch logarithmic graphs, and how to apply transformations. The article also covers the laws of logarithms, the change of base formula, and solving logarithmic equations. Includes step-by-step worked examples, DSE exam techniques, practice questions. These are high-level skills that distinguish top-performing students.
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Logarithmic Functions | Definition, Properties & Graphs
Question 1 MC
Find the domain of \( f(x) = \log_4 (3-x) \).
A. \( x > 3 \) B. \( x < 3 \) C. \( x \ge 3 \) D. \( x \le 3 \)
Question 2 MC
What is the asymptote of \( y = \log_3 x - 2 \)?
A. \( x = 0 \) B. \( x = 2 \) C. \( y = -2 \) D. \( y = 0 \)
Question 3 Short Answer
Simplify \( \log_5 125 - \log_5 5 \).
Question 4 Short Answer
Write \( \log_2 16 + \log_2 4 \) as a single logarithm.
Question 5 MC
Solve \( \log_3 x = 4 \).
A. \( 12 \) B. \( 27 \) C. \( 81 \) D. \( 64 \)
Question 6 Short Answer
Use change of base to evaluate \( \log_2 7 \) (to 3 decimal places).
Question 7 MC
Which transformation shifts \( y = \log_2 x \) right by 3 units?
A. \( y = \log_2 (x+3) \) B. \( y = \log_2 (x-3) \) C. \( y = \log_2 x + 3 \) D. \( y = \log_2 x - 3 \)
Question 8 Short Answer
Solve \( \log_2 (x-1) = 3 \).
Question 9 Short Answer
If \( \log_a 64 = 3 \), find \( a \).
Question 10 Short Answer
Sketch \( y = \log_3 (x+1) \), labeling the asymptote and intercept.
When solving logarithmic equations, always check that the solution is within the domain (argument > 0). Extraneous solutions can arise from squaring or other operations.