Prior Knowledge
Inverse proportionality (MA5.2-5NA) (NESA, 2012)
Inverse relationships - Relevant when exploring the inverse relationship between exponentials and logarithms.
Calculus (MA-C1)(NESA, 2017)
Different strategies of differentiation - Students will be required to differentiate functions in this unit that will require a variety of methods in order to solve.
The gradient function - This is especially critical when explaining the principle behind d(e^x)/dx = e^x
Indices (ACMNA149)(ACMNA150)(ACMNA182) (NESA, 2012)
Exponents/Law of indices - Students will be interacting with exponents and manipulating functions using the law of indices as part of simplifying functions. This is a really big part of this unit, especially when working with functions/expressions involving e^x.
Logarithms and Exponentials (Introducing logarithms)(Logarithmic laws and applications) (NESA, 2012)
An understanding of logarithms as a function and logarithm laws
Features of the logarithm function learned in earlier lessons and it relationships with the exponential function
Logarithms and Exponentials (Work with natural logarithms in a variety of practical and abstract contexts AAM) (ACMMM155)
An understanding of logarithm laws and solving equations involving indices using logarithms
An understanding of the exponential function
It is also important to note that content learnt throughout the unit, especially at the start, will become prior knowledge for later lessons in the unit.
Misconceptions
Inverse relationship between exponentials and logarithms
Challenge when working with logs (Hewson, 2013):
Challenges when converting between logarithmic to exponential form
Difficulty thinking of indices as logarithms instead of a simple number
Logarithmic Scales
Common misconceptions revolve around the reading and interpreting of logarithmic scales (Menge et al., 2018). These can include:
Students incorrectly assume that the gaps between values on the log scale are equivalent to each other.
Difficulty determining the scale between values not next to each other on a log scale.
Differentiating e^x
An unclear grasp of differentiation rules (Othman et al., 2018). This can result in challenges such as:
Students incorrectly differentiating e^x as expressions such as x*e^(x-1), x*e or e.
When working with constants k and a in f(x) = k*e^(a*x), students may be confused as to how these are impacted when differentiated.
Logarithm Laws and the logarithm function
Misconceptions with logarithm laws, treating log as a variable whereas the other is by possibly participants thinking log(x)/log(y) = log(x/y). (Liang and Wood, 2005)
Students struggle with the topic of logarithms in general. (Hewson, 2013).
Methods for solving equations involving indices
Students have misconceptions about logarithm laws and the methods involved for solving equations involving indices, with the incorrect and improper use of laws in logarithms (Ganesan and Dindyal)