Ch 2 · 2.2 / 3.2 Logarithmic and Exponential Functions · A Level Mathematics 9709
Topic 2 of 9 in Pure Mathematics 2 and 3 — A Level Mathematics 9709 — 55 lessons.
Introduction
Bacteria doubling every hour, a cup of tea cooling, money earning compound interest: these grow or decay exponentially. Logarithms undo exponentials, so they are how you find "how long until…". They also turn curved data into straight lines, which is how scientists find the constants in a law. This content is identical for Paper 2 (section 2.2) and Paper 3 (section 3.2).
Key ideas — Logarithms and indices
log_a y = x iff a^x = y (a > 0,\ a ≠ 1,\ y > 0)
🧑🏫 Teacher's explanation
Think of logarithms as a 'reverse' operation, much like subtraction is the reverse of addition. If an index (exponent) asks, "What do I get when I raise a to the power of x?", the logarithm asks, "To what power must I raise a to get y?" It is essentially a way to isolate an exponent.
📖 Definitions
🔍 Worked example — Easy
Solve for x: 5^x-1 = 125
🔍 Worked example — Standard
Show that the relationship y = kx^n can be written in the linear form ln y = n ln x + ln k.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — e and the natural logarithm
e ≈ 2.718 is the base for which y = e^x has gradient equal to y at every point. The natural logarithm ln x = log_e x is its inverse:
🧑🏫 Teacher's explanation
Imagine you are tracking the growth of a population of bacteria. If the population grows at a rate proportional to its current size, the time it takes to reach a certain size is best described by the number e ≈ 2.718. This number is special because the function y = e^x is its own derivative; in simple terms, the 'speed' of growth at any moment is exactly equal to the 'amount' present at that moment.
📖 Definitions
🔍 Worked example — Easy
Solve for x: e^2x-3 = 5
🔍 Worked example — Standard
Show that the graph of y = e^kx is a reflection of y = e^x in the y-axis when k = -1.
🔍 Worked example — Exam-style
A relationship is given by y = k(2^x). By considering ln y, express ln y in the form mx + c and state the values of m and c in terms of k.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Solving equations with the unknown in a power
Take logs of both sides (usually ln) and use ln a^x = xln a:
🧑🏫 Teacher's explanation
When you see a variable stuck up in a power, like 3^x, it is essentially 'locked' away from the rest of the equation. To get it down to ground level where we can solve for it, we need a mathematical 'key'. That key is the logarithm. Because logarithms are the inverse of exponential functions, applying a logarithm to both sides of an equation 'undoes' the exponentiation.
📖 Definitions
🔍 Worked example — Easy
Solve 5^x+1 = 12
🔍 Worked example — Standard
Solve e^2x - 5e^x + 6 = 0
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Inequalities
Solve as for equations, but remember: dividing by a negative number reverses the inequality, and ln a < 0 when 0 < a < 1:
🧑🏫 Teacher's explanation
Solving inequalities involving exponents is almost identical to solving standard linear inequalities, with one massive 'trap' to watch out for. When you take the natural logarithm (ln) of both sides to bring the variable down from the exponent, you are essentially applying a function to both sides. If that function is increasing, the inequality sign stays the same. If it is decreasing, the sign flips.
📖 Definitions
🔍 Worked example — Easy
Solve the inequality 2^x > 16.
🔍 Worked example — Exam-style
Solve the inequality 3^2x-1 < 5^x+2.
⚠️ Common errors
🧠 Quick check 1
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🎯 Exam question
Key ideas — Reducing a law to a straight line
Find the gradient from two points on the line, then the intercept, then undo the logs.
🧑🏫 Teacher's explanation
In many scientific experiments, we encounter relationships that aren't straight lines, such as power laws like y = kx^n or exponential growth like y = ka^x. These are difficult to analyze directly because we can't easily find the 'slope' of a curve. However, if we apply logarithms to both sides of the equation, we can 'straighten' the relationship into a linear form (Y = mX + c). Once the line is straight, we can use simple linear regression techniques—finding the gradient and intercept—to find the original constants k and n or a.
📖 Definitions
🔍 Worked example — Easy
A relationship is given by y = kx^3. By taking the natural logarithm of both sides, show that a plot of ln y against ln x yields a straight line and identify its gradient.
🔍 Worked example — Standard
The relationship between temperature T and time t is given by T = k(2^t). A student plots ln T against t and finds the line passes through (0, 0.693) and (2, 2.079). Find the value of k.
🔍 Worked example — Exam-style
A curve follows the law y = kx^n. A graph of ln y against ln x is drawn. The line passes through (1, 5) and (3, 11). Find the values of n and k.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
(a) Show that a graph of ln P against ln V is a straight line and state its gradient. [2] (b) A graph of ln P against ln V passes through the points (2, -1.2) and (4, -2.5). Find the values of k and the gradient. [4] [6 marks]
Worked examples
Solve 3^x+1 = 4^2x-1.
🔍 Worked example — Easy
Solve 3^x+1 = 4^2x-1
🔍 Worked example — Standard
Solve e^2x - 5e^x + 6 = 0
🔍 Worked example — Exam-style
The graph of ln y against ln x is a straight line passing through (0.5, 2.1) and (2.0, 4.5). Find the constants k and n for the relationship y = kx^n.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) x = ln 40/ln 5 = 2.29. (2) lnx^2/x + 6 = 0 ⇒ x^2 = x + 6 ⇒ x = 3 (reject -2). (3) xln 3 > (x + 3)ln 2 ⇒ x > 3ln 2/ln 1.5 = 5.13. (4) (2e^x + 3)(e^x - 1) = 0; e^x = -tfrac32 impossible, so x = 0. (5) n = -0.5; ln k = 1 + 0.5 × 2 = 2, k = e^2 = 7.39.