Ch 6 · 6 Logarithmic and Exponential Functions · O Level Additional Mathematics 4037
Topic 6 of 6 in Algebra and Functions — O Level Additional Mathematics 4037 — 63 lessons.
Introduction
Earthquake magnitudes, the loudness of sound in decibels and the pH of a solution are all logarithmic scales, because the quantities they measure grow exponentially. A logarithm answers "what power?": log_2 8 = 3 because 2^3 = 8. This topic covers the laws of logs, the graphs of e^x and ln x, and how to solve equations with the unknown in a power. (Syllabus topic 6.)
Key ideas — Logarithms are powers
log_a b = c iff a^c = b (a > 0,\ a ≠ 1,\ b > 0)
🧑🏫 Teacher's explanation
Think of a logarithm as a 'detective' looking for a missing exponent. If you see the equation 2^x = 8, you know the answer is 3 because 2 raised to the power of 3 is 8. A logarithm simply formalizes this relationship. Instead of asking "What is 2 to the power of 3?", we ask "To what power must we raise 2 to get 8?", which we write as log_2 8 = 3.
📖 Definitions
🔍 Worked example — Easy
Evaluate log_3 81.
🔍 Worked example — Standard
Solve for x: 5^x-1 = 125.
🔍 Worked example — Exam-style
Show that log_2 10 = 1 + log_2 5.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — The laws of logarithms
log_a xy = log_a x + log_a y log_ax/y = log_a x - log_a y log_a x^n = nlog_a x
🧑🏫 Teacher's explanation
Think of logarithms as a 'translator' that turns multiplication into addition, division into subtraction, and powers into simple multiplication. If you think of exponents as the 'action' of growing a number by repeated multiplication, the logarithm is the 'question' asking how many times you used that base to reach a certain result.
📖 Definitions
🔍 Worked example — Easy
Simplify log_2 8 + log_2 4
🔍 Worked example — Standard
Write 2log_3 x - log_3 y as a single logarithm.
🔍 Worked example — Exam-style
Evaluate log_25 5 using the change of base formula.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Graphs of exponentials and logs
🧑🏫 Teacher's explanation
Think of exponential and logarithmic functions as two sides of the same coin. If an exponential function tells you how fast something grows (like interest in a bank account), the logarithmic function tells you how much time you need to reach a certain amount. Because they do opposite jobs, they are 'inverses'. Imagine looking at the graph of y = e^x in a mirror placed along the diagonal line y = x; the reflection you see is y = ln x.
📖 Definitions
🔍 Worked example — Easy
Sketch the graph of y = 2e^3x - 4 and state the equation of its horizontal asymptote.
🔍 Worked example — Standard
Find the equation of the vertical asymptote for the graph y = 3ln(2x - 6) + 1.
🔍 Worked example — Exam-style
A function is defined as f(x) = e^2x + 3. Find the value of x for which f(x) = 11.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Key ideas — Solving aˣ = b
Take logs of both sides: xlg a = lg b, so x = lg b/lg a (or use ln). For harder equations, use the laws to combine logs into one, or substitute to make a quadratic (topic 4): 2^2x - 5(2^x) + 4 = 0 is u^2 - 5u + 4 = 0 with u = 2^x.
🧑🏫 Teacher's explanation
Think of an exponential equation like a locked box where the unknown x is hidden in the 'power' position. If you have 2^x = 10, you can't simply divide by 2 because the 2 is not being multiplied; it is the base of a power. To 'unlock' the x, we need a mathematical tool that specifically looks at exponents. This tool is the logarithm.
📖 Definitions
🔍 Worked example — Easy
Solve 5^x = 12
🔍 Worked example — Standard
Solve 3^2x-1 = 27^x
🔍 Worked example — Exam-style
Solve 4^x - 6(2^x) + 8 = 0
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples — Example 1: sketching with asymptotes
y = 3e^2x - 4: asymptote y = -4; y-intercept 3 - 4 = -1; crosses the x-axis where e^2x = tfrac43, i.e. x = tfrac12lntfrac43 = 0.144.
🔍 Worked example — Easy
Sketch the graph of y = 3e^2x - 4, stating the equation of the asymptote and the y-intercept.
🔍 Worked example — Standard
Sketch the graph of y = 2ln(x + 3), stating the equation of the asymptote and the x-intercept.
🔍 Worked example — Exam-style
A curve has the equation y = kln(x - 2) + 5. The curve passes through the point (3, 7). Find the value of k and state the equation of the vertical asymptote.
⚠️ Common errors
Worked examples — Example 2: change of base
Solve log_3 x + log_9 x = 6. Since log_9 x = tfrac12log_3 x: tfrac32log_3 x = 6, so log_3 x = 4 and x = 81.
🔍 Worked example — Easy
Simplify log_5 20 + log_5 50 into a single logarithm.
🔍 Worked example — Standard
Solve log_2 x + log_4 x = 6.
🔍 Worked example — Exam-style
Solve log_x 3 + log_2 3 = 2.
⚠️ Common errors
Worked examples — Example 3: combining logs
Solve log_2(x + 1) - log_2(x - 2) = 2. log_2x + 1/x - 2 = 2 ⇒ x + 1/x - 2 = 4 ⇒ x + 1 = 4x - 8 ⇒ x = 3 (valid: both logs defined).
🔍 Worked example — Easy
Simplify log_3 54 - log_3 2 into a single logarithm.
🔍 Worked example — Standard
Solve log_5(x) + log_5(x - 4) = 2
🔍 Worked example — Exam-style
Solve log_2(x + 1) - log_2(x - 2) = 2
⚠️ Common errors
Worked examples — Example 4: unknown in a power
5^x = 20 ⇒ x = lg 20/lg 5 = 1.86. And 2^2x - 5(2^x) + 4 = 0 ⇒ (u - 1)(u - 4) = 0, so 2^x = 1 or 4, giving x = 0 or x = 2.
🔍 Worked example — Easy
Solve 3^x = 50. Give your answer correct to 3 decimal places.
🔍 Worked example — Standard
Solve 2^2x - 5(2^x) + 4 = 0.
🔍 Worked example — Exam-style
Solve 5^x+1 = 12.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) log_a9 × 4/6 = log_a 6. (2) lg2x + 1/x - 4 = 1 ⇒ 2x + 1 = 10x - 40 ⇒ x = 41/8. (3) (2x + 1)lg 3 = lg 7 ⇒ x = tfrac12 (lg 7/lg 3 - 1 ) = 0.386. (4) asymptote y = 2; y-intercept 1; x-intercept where e^-x = 2, x = -ln 2. (5) log_2 p = log_4 p/log_4 2 = 2q; log_p 16 = log_4 16/log_4 p = 2/q.