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06062025–2027 syllabus

ADDITIONAL MATHEMATICS · CHAPTER 6

Logarithmic and exponential functions

Use inverse relationships, laws and domains to solve growth-style equations and interpret transformed graphs.

Additional Mathematics4 connected sectionsSyllabus-aligned guide

LEARNING OBJECTIVES

What you will be able to do

  • interpret logarithms as inverse exponents
  • apply logarithm laws and change of base
  • sketch exponential and logarithmic graphs with asymptotes
  • solve equations of the form aˣ = b and related forms

AT A GLANCE

Syllabus0606Coverage2025–2027Sections4LevelAdditional Mathematics

INTRODUCTION · THE BIG IDEA

Use inverse relationships, laws and domains to solve growth-style equations and interpret transformed graphs.

A logarithm answers an exponent question: logₐb is the power to which a must be raised to produce b. This makes logarithms the natural tool for bringing an unknown exponent down into an equation.

Every real logarithm requires a positive argument. This domain condition is essential when combining logarithms or checking solutions.

01

SECTION 01

Inverse functions and graph behaviour

Core concept

The functions eˣ and ln x are inverses, so their graphs reflect in y = x. Exponential graphs have horizontal asymptotes; logarithmic graphs have vertical asymptotes.

Typical linear, quadratic, cubic, reciprocal and exponential graph shapeslinearquadraticcubicreciprocalexponential
Original graph-family guide: use intercepts, symmetry, turning points and asymptotes to identify each family.
Parent graphs
FunctionDomainRangeAsymptote
y = eˣall real xy > 0y = 0
y = ln xx > 0all real yx = 0
y = keⁿˣ + aall real xdepends on k and ay = a
y = k ln(ax + b)ax + b > 0all real y if k ≠ 0x = −b/a
02

SECTION 02

Laws of logarithms

Core concept

The laws come from index rules: multiplying powers adds exponents, dividing subtracts them, and raising a power multiplies exponents.

RULE 1
logₐ(MN) = logₐM + logₐN
RULE 2
logₐ(M/N) = logₐM − logₐN
RULE 3
logₐ(Mᵏ) = k logₐM
RULE 4
logₐM = log_bM / log_ba
Original worked example

Condense and solve a logarithmic equation

  1. Solve log₂(x − 1) + log₂(x + 2) = 3.
  2. Require x − 1 > 0 and x + 2 > 0, so x > 1.
  3. Combine: log₂[(x − 1)(x + 2)] = 3.
  4. Convert: (x − 1)(x + 2) = 8, giving x² + x − 10 = 0.
  5. The roots are (−1 ± √41)/2; keep only the root greater than 1.

Answer: x = (−1 + √41)/2.

03

SECTION 03

Exponential equations

Core concept

Rewrite both sides with a common base when possible. Otherwise take logarithms and divide by the coefficient of x.

RULE 1
aˣ = b ⇒ x = logₐb = ln b / ln a
RULE 2
eᵘ = v ⇒ u = ln v, for v > 0
ORIGINAL STUDY DIAGRAMSolve an exponential equation
1Isolate the exponential term
2Check its required sign
3Use a common base or take logs
4Solve the linear exponent
5Check in the original
04

SECTION 04

Transformations and restrictions

Core concept

A vertical translation changes the horizontal asymptote of an exponential graph. Inside a logarithm, ax + b shifts the vertical asymptote to ax + b = 0 and restricts the valid side.

DETAILED EXPLANATION

  • State asymptote equations, not only their direction.
  • The sign of k reflects a graph vertically.
  • The sign of n controls exponential increase or decrease.
  • Reject any result that makes a log argument zero or negative.

QUICK CHAPTER SUMMARY

The ideas to carry forward

  • Logarithmic and exponential functions are inverses.
  • Log arguments must be positive.
  • Log laws mirror index laws.
  • Asymptotes are part of a complete graph description.
  • Unknown exponents can be solved exactly with logarithms.

QUICK REVISION CHECKLIST

Can you do each of these without your notes?

  • interpret logarithms as inverse exponents
  • apply logarithm laws and change of base
  • sketch exponential and logarithmic graphs with asymptotes
  • solve equations of the form aˣ = b and related forms
← Simultaneous equationsStraight-line graphs →