Exponential and logarithmic functions
The laws of logarithms, solving exponential and logarithmic equations, and their graphs. The unit is NF, but it is essential for students aiming at level 4 or above.
What you will learn
- 3.1–3.2 Rational indices: a^(1/n) = ⁿ√a, a^(m/n) = (ⁿ√a)^m; the index laws.
- 3.3 Logarithms and their properties, including change of base.
- 3.4 Graphs of y = aˣ and y = logₐx: domains, increasing or decreasing, symmetry about y = x, intercepts.
- 3.5 Exponential and logarithmic equations. Equations that turn into quadratics (such as 4ˣ − 3·2ˣ − 4 = 0) belong to F5 more about equations.
- 3.6 Applications: Richter scale, decibels.
- 3.7 History of logarithms (appreciation only).
You need first: Indices and scientific notation, Functions and graphs
Key terms
| English | 中文 |
|---|---|
| rational index | 有理指數 |
| exponential function | 指數函數 |
| logarithm, base | 對數，底 |
| common logarithm (log) | 常用對數 |
| logarithmic function | 對數函數 |
| change of base formula | 換底公式 |
| exponential equation | 指數方程 |
| logarithmic equation | 對數方程 |
Standard steps
The one idea
log is the reverse of a power. "logₐN = x" means "aˣ = N". So log₂32 asks "2 to what power is 32?" Answer: 5.
Properties
| Rule | Why |
|---|---|
| logₐ1 = 0, logₐa = 1 | a⁰ = 1, a¹ = a |
| log MN = log M + log N | multiplying powers adds the indices |
| log (M/N) = log M − log N | dividing subtracts |
| log Mᵏ = k log M | (aˣ)ᵏ = a^(kx) |
| log_b N = logₐN / logₐb | change of base, so the calculator's log (base 10) can be used |
Not rules: log(M + N) ≠ log M + log N; (log M)/(log N) ≠ log(M/N).
- Write each term as the log of a single number or letter, with any multiplier in front.
- Apply one rule from the table per line: move multipliers up as powers, then combine sums and differences.
- Check you have not used one of the "not" rules above.
Solve aˣ = N (x in the index)
- Take log of both sides: x log a = log N.
- x = log N ÷ log a.
- Use the calculator and round only at the end, usually to 3 significant figures.
Solve a log equation
- Use the rules to make a single log on each side.
- Remove the log: logₐ(stuff) = c becomes stuff = aᶜ.
- Check each answer: the inside of every log must be positive in the original equation.
Graphs
- y = aˣ (a > 1): passes through (0, 1), always above the x-axis, increases.
- y = logₐx (a > 1): passes through (1, 0), only for x > 0, increases.
- They are reflections of each other in y = x.
- For 0 < a < 1 both decrease.
Worked examples
= (³√8)² = 2² = 4.
(x + 1) log 3 = log 20
x + 1 = log 20 / log 3 ≈ 2.7268
x ≈ 1.73x + 1 = 2³ = 8
x = 7 check: log₂8 = 3 ✓log 8 / log 4 = 3 log 2 / (2 log 2) = 3/2.
2 log 5 = log 5² = log 25
log 12 − log 3 + log 25 = log (12 ÷ 3 × 25)
= log 100
= 2Common mistakes
- log(x + 3) "expanded" into log x + log 3.
- log 8 / log 4 simplified as log 2.
- Forgetting to reject an answer that makes the inside of a log zero or negative.
- In 3^(x+1), dividing by log 3 before expanding, and losing the bracket.
- Reading y = aˣ as crossing the x-axis (it never does).
How the DSE tests it
- Section B: log equations, often turned into a quadratic (F5 skill), or log simultaneous equations.
- Paper 2 Section B: rational index simplification; log rules; graphs of y = aˣ and y = logₐx; a linear relation between log y and x read from a straight-line graph.
- Real-life: a Richter scale or decibel question with the formula given.
Questions you might not ask in class
What is a log actually for?
It answers "what power?". Without logs you can't solve 3ˣ = 20, because x is stuck in the index.
Why must I check the answer?
Squaring and removing logs can create answers that don't work in the original equation. A log of a negative number does not exist.
Which log does the calculator use?
"log" is base 10 and "ln" is base e. In this unit, change of base lets either one work, as long as you use the same one on top and bottom.