Outcomes
A student:

 MA5.31WM
uses and interprets formal definitions and generalisations when explaining solutions and/or conjectures

 MA5.33WM
uses deductive reasoning in presenting arguments and formal proofs

 MA5.311NA
uses the definition of a logarithm to establish and apply the laws of logarithms
Content
 Students:
 Use the definition of a logarithm to establish and apply the laws of logarithms (ACMNA265)
 define 'logarithm': the logarithm of a number to any positive base is the index when the number is expressed as a power of the base, ie \( a^x = y \Leftrightarrow \log_a y = x \) where \(a>0\), \(y>0\)

translate statements expressing a number in index form into equivalent statements expressing the logarithm of the number, eg
\( \begin{align} 9 &= 3^2, & \therefore & \,\,\log_{3}9 = 2 \\ \frac{1}{2} &= 2^{1}, & \therefore & \,\,\log_{2}\frac{1}{2} = 1\\ 4^{\frac{3}{2}} &= 8, & \therefore\, & \,\,\log_{4}8 = \frac{3}{2} \end{align} \) 
deduce the following laws of logarithms from the laws of indices:
\( \begin{align} \log_a x + \log_a y &= \log_a (xy) \\ \log_a x  \log_a y &= \log_a \left(\frac{x}{y}\right) \\ \log_a x^n &= n\log_a x \end{align} \)

establish and use the following results:
\( \begin{align} \log_a a^x &= x \\ \log_a a &= 1 \\ \log_a 1 &= 0\\ \log_a \left(\frac{1}{x}\right) &= \log_a x \end{align} \)
 apply the laws of logarithms to simplify simple expressions, eg \( \log_2 8 \), \( \log_{81} 3 \), \( \log_{10} 25 + \log_{10} 4 \), \( 3\log_{10} 2 + \log_{10} (12.5) \), \( \log_2 18  2\log_2 3 \)
 simplify expressions using the laws of logarithms, eg simplify \( 5 \log_a a  \log_a a^4 \)
 draw and compare the graphs of the inverse functions \( y= a^x \) and \( y = \log_a x \)
 relate logarithms to practical scales, eg Richter, decibel and pH scales (Problem Solving)
 compare and contrast exponential and logarithmic graphs drawn on the same axes, eg \( y=2^x \), \( y=\log_2 x \), \( y = 3^x \), \( y = \log_3 x \) (Communicating, Reasoning)
 Solve simple exponential equations (ACMNA270)

solve simple equations that involve exponents or logarithms,
eg \( 2^t = 8 \), \( 4^{t+1} = \frac{1}{8\sqrt{2}} \), \( \log_{27} 3 = x \), \( \log_4 x = 2 \)
Background Information
Logarithm tables were used to assist with calculations before the use of handheld calculators. They converted calculations involving multiplication and division to calculations involving addition and subtraction, thus simplifying the calculations.
Purpose/Relevance of Substrand
The laws obeyed by logarithms make them very useful in calculation and problem solving. They are used widely in pure mathematics, including calculus, and have many applications in engineering and science, including computer science. In acoustics, for example, sound intensity is measured in logarithmic units (decibels), and in chemistry 'pH' is a logarithmic measure of the acidity of a solution. Logarithms are common in mathematical and scientific formulas and are fundamental to the solution of exponential equations.
Language
Teachers need to emphasise the correct language used in connection with logarithms,
eg \(\log_aa^x=x\) is 'log to the base \(a\), of \(a\) to the power of \(x\), equals \(x\)'.