Students begin by developing a clear understanding of exponential relationships and how they differ from linear and polynomial patterns. They explore the graphs, transformations, and key features of exponential functions while learning to distinguish between exponential growth and decay. The unit concludes with applications involving population growth, depreciation, compound interest, and radioactive decay.
This unit introduces logarithms as the inverse of exponential functions. Students learn to convert between exponential and logarithmic forms, graph logarithmic functions, and apply the product, quotient, and power properties of logarithms. They also investigate common and natural logarithms and examine how logarithmic scales are used to represent quantities with very large ranges of values.
Students develop techniques for solving exponential and logarithmic equations and inequalities. They solve equations by rewriting expressions with common bases, applying logarithms, combining logarithmic terms, and using graphical methods. Particular attention is given to domain restrictions, extraneous solutions, verification, and selecting efficient solution strategies.
In this unit, students investigate more advanced models involving the natural base
e, natural logarithms, continuous growth and decay, and logistic functions. They compare discrete and continuous models, interpret carrying capacity, and determine which mathematical models are most appropriate for different situations. Emphasis is placed on connecting equations, graphs, data, and real-world meaning.
The final unit integrates concepts from throughout the course through applications in finance, science, technology, and advanced mathematics. Students use exponential and logarithmic models to analyse investments, loans, scientific measurements, decay processes, population change, and technological trends. They also explore how exponential and logarithmic functions connect to rates of change and prepare students for differentiation and integration in calculus.
Subtopics
Finance Applications
Scientific Applications
Exponential Models in Technology
Connections to Calculus
Comprehensive Problem Solving
By the end of this course, students will be able to represent and analyse exponential and logarithmic functions, solve equations and inequalities, construct and evaluate mathematical models, interpret growth and decay in context, and communicate complete mathematical reasoning. They will understand the inverse relationship between exponential and logarithmic functions and recognise their importance across mathematics, science, finance, technology, and engineering.