11–12Mathematics Advanced 11–12 Syllabus (2024)

Implementation from 2026

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Content

Year 11

Teaching advice

Examples

Exponential and logarithmic functions

MAO-WM-01
develops understanding and fluency in mathematics through exploring and connecting mathematical concepts, choosing and applying mathematical techniques to solve problems, and communicating their thinking and reasoning coherently and clearly
MAV-11-07
applies exponential and logarithmic laws to simplify expressions, solve equations and prove results
MAV-11-08
analyses graphs of exponential and logarithmic functions

Exponential and logarithmic functions

Exponential functions

Graph the exponential functions $y = k (a^{x})$ and $y = k (a^{- x})$ for constants $a$ and $k$ where $a > 0$ , $a \neq 1$ and $k \neq 0$ , and identify its asymptote, $y$ -intercept, domain and range
Describe the behaviour of $y = k (a^{x})$ and $y = k (a^{- x})$ as $x \to \infty$ and $x \to - \infty$
Examine the gradient of the tangent to the curve $y = a^{x}$ at its $y$ -intercept for varying values of $a$ , and verify using graphing applications that there is a unique number $e \approx 2.71828182845$ ..., such that the gradient of the tangent to $y = e^{x}$ at $x = 0$ is 1, and call this number $e$ Euler’s number
Examine the gradient function of $y = a^{x}$ with graphing applications and identify that the gradient function is $\frac{d}{dx} (a^{x}) = k a^{x}$ , for some constant $k$ depending only on $a$
Conclude that Euler’s number, $e$ , is a unique number such that $\frac{d}{dx} (e^{x}) = e^{x}$ , that is, $e^{x}$ is its own derivative

Logarithmic functions

Define the logarithm of a number $y$ , where $y > 0$ , to any positive base $a$ as the index to which $a$ is raised to give $y$
Use the notation $\log_{a} y$ for the logarithm of $y$ to the base $a$
Define the natural logarithm $\ln a = \log_{e} a$
Use digital tools to determine Loading and Loading values of exponential and logarithmic Loading
Explain that $y = a^{x}$ is equivalent to $x = \log_{a} y$ for $a > 0$ and $a \neq 1$ , and use the equivalence to solve equations of the form $a^{x} = b$ , for $a > 0$
Recognise and use the logarithmic properties: $\log_{a} a^{x} = x$ for all real $x$ , $a^{\log_{a} x} = x$ where $x > 0$
Derive the laws of logarithms from the laws of indices $\log_{a} m + \log_{a} n = \log_{a} (mn)$ , $\log_{a} m - \log_{a} n = \log_{a} (\frac{m}{n})$ and $\log_{a} (m^{n}) = n \log_{a} m$
Justify the logarithmic results $\log_{a} a = 1$ , $\log_{a} 1 = 0$ and $\log_{a} (\frac{1}{x}) = - \log_{a} x$
Apply the logarithmic laws and results to simplify expressions, solve equations and prove results, using digital tools where necessary
Prove the change of base rule $\log_{a} x = \frac{\log_{b} x}{\log_{b} a}$ and use it to solve problems
Graph the logarithmic function $y = \log_{a} x$ for $a > 0$ and $a \neq 1$
Use graphing applications to identify the graphs of $y = a^{x}$ and $y = \log_{a} x$ as reflections of each other in the line $y = x$ , in cases where $a > 1$ and $0 < a < 1$

Related files

Year 12