\(y=a^x\)
Compare bases above and below 1, transformations, asymptotes, growth and decay.
IB MATHEMATICS AA · TOPIC 2 FUNCTIONS · PREMIUM
Exponents, Exponential Equations & Logarithms
This chapter builds exponentials and logarithms as a story rather than a collection of rules. Students first understand what repeated multiplication means, then see how the graph \(y=a^x\) changes with the base, why logarithms were needed as the inverse language of exponentials, and how exact algebra, change of base and GDC methods fit together.
THE STORY OF THE CHAPTER
Exponentials multiply. Logarithms undo that multiplication.
The notes begin with powers as repeated multiplication and then let the exponent become a variable. That creates the family \(y=a^x\). Students compare growth and decay for different values of a, study transformations and asymptotes, and only then introduce logarithms as the inverse operation needed to answer the question: what exponent produced this number?
\(\log_a b=c\)
Understand the inverse relationship \(a^c=b\) before memorising any laws.
One common language
Turn changing or awkward logarithm bases into a single base and expose hidden algebra.
Structure before buttons
Use graph intersections only when exact algebra stops, then progress into parameters and linked synthesis.
A CENTRAL DECISION
Match bases before reaching for logarithms.
Many exponential equations are designed so that apparently different numbers share a common base. For example, powers of 4 and 8 can both be rewritten in base 2. Once both sides are written as powers of the same positive base, equality of the powers means equality of the exponents. If the bases genuinely cannot be matched, logarithms become the tool that brings the exponent down into ordinary algebra.
\[8=2^3\qquad 4=2^2\qquad 27=3^3\qquad 9=3^2\]
The worksheet repeatedly tests this decision so students learn to choose a method rather than mechanically take logarithms every time.
GDC + EXACT ALGEBRA
The calculator is used only when the mathematics says it should be.
The pack contains exact no-GDC equations, cubic exponential substitutions, logarithmic domain checks, changing bases, reciprocal powers and parameter questions. When equations mix exponential and algebraic behaviour so that an exact elementary solution is unavailable, the notes show how to graph both sides and find every intersection.
INSIDE THE COMPLETE PACK
Detailed teaching plus serious practice.
38 pages
Concise narrative explanations, graph-value tables, exact exponential and logarithmic equations, polynomial long division, linked IB-style examples, and detailed Casio/TI-Nspire GDC routes.
43 questions · 13 pages
Original CF Maths problems, simultaneous systems, exact no-GDC questions, harder logarithmic equations, past-paper reference structures and a long Paper 3-style investigation.
14 pages
Rebuilt solutions matching the revised worksheet, including domain checks, exact answers, parameter cases, numerical intersections and the full final investigation.
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Included in Topic 2 · Functions · €50
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Exponents & Logarithms 60-minute diagnostic · €7
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The Composition & Inverse Functions pack develops the inverse-function idea visually and algebraically, which makes the later exponential/logarithm relationship much more natural.