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O-Level Exponential and logarithmic functions
What the O-Level syllabus expects for Exponential and logarithmic functions, and how to practise it.
What the syllabus expects
- Understanding the functions a^x, e^x, loga x and ln x with their graphs, taking in the laws of logarithms, the equivalence between y = a^x and x = loga y, and switching the base of a logarithm
- Reducing expressions and solving straightforward equations that involve exponentials and logarithms
- Modelling with exponential and logarithmic functions
How it's examined
Questions on this topic most often ask you to find, solve, compare, evaluate. About 9% of the past-paper style questions in Rae's bank for this subject sit in this topic.
Worked examples
Example 1 (4 marks)
Find the solution of the equation 2e^x = 3 - 5√(e^x).
Show the worked answer
2e^x = 3 - 5*sqrt(e^x) Let u = sqrt(e^x), so u > 0 and e^x = u². 2u² = 3 - 5u 2u² + 5u - 3 = 0 (2u - 1)(u + 3) = 0 u = 1/2 or u = -3 Since u = sqrt(e^x) > 0, reject u = -3, so u = 1/2. sqrt(e^x) = 1/2 => e^x = 1/4 x = ln(1/4) = -2 ln 2 (approx -1.386)
Example 2 (3 marks)
If log₂(y²) = 4 - log₀.₅ x, write y as a function of x.
Show the worked answer
Since 0.5 = 2⁻¹, log_0.5 x = log_2 x / log_2(0.5) = log_2 x / (-1) = -log_2 x. So log_2(y²) = 4 - (-log_2 x) = 4 + log_2 x = log_2 16 + log_2 x = log_2(16x). Hence y² = 16x, and taking the positive root y = 4 sqrt(x).
Example 3 (3 marks)
For the cooling cube with T = 30 + 175e^(-mt), the temperature is 128°C at t = 3. Find the value of m.
Show the worked answer
T = 30 + 175 e^(-mt), with T = 128 at t = 3: 128 = 30 + 175 e^(-3m) 98 = 175 e^(-3m) e^(-3m) = 98/175 = 0.56 -3m = ln(0.56) = -0.5798 m = 0.5798/3 = 0.193.
More worked questions on this topic
- Bacteria grow by N = a·10^(kt), t hours after the start. The data are t = 2,4,6,8 with N = 900, (4 marks)
- A radioactive sample’s mass y mg falls as time x hours passes, and an analyst proposes the mode (4 marks)
- Explain why, according to T = 30 + 175e^(-mt), the cube’s temperature can never drop below 30°C (2 marks)
- For two quantities R and t the recorded pairs (t, R) are (20, 2.79), (10, 2.32), (40, 4.74), (3 (2 marks)
More O-Level Additional Maths (A-Maths) topics
Quadratic functions · Equations and inequalities · Surds · Polynomials and partial fractions · Binomial expansions · Trigonometric functions, identities and equations · all of O-Level Additional Maths (A-Maths)