Logarithmic functions:
define logarithms as indices: \(a^x=b\) is equivalent to \(x=\log_ab\) i.e. \(a^{\log_ab}=b\)
(ACMMM151)
establish and use the algebraic properties of logarithms
(ACMMM152)
recognise the inverse relationship between logarithms and exponentials: \(y=a^x\) is equivalent to \(x=\log_ay\)
(ACMMM153)
interpret and use logarithmic scales such as decibels in acoustics, the Richter Scale for earthquake magnitude, octaves in music, pH in chemistry
(ACMMM154)
solve equations involving indices using logarithms
(ACMMM155)
recognise the qualitative features of the graph of \(y=\log_ax\) \((a>1)\) including asymptotes, and of its translations \(y=\log_ax+b\) and \(y=\log_a{(x+c)}\)
(ACMMM156)
solve simple equations involving logarithmic functions algebraically and graphically
(ACMMM157)
identify contexts suitable for modelling by logarithmic functions and use them to solve practical problems.
(ACMMM158)
Calculus of logarithmic functions:
define the natural logarithm \(\ln x=\log_ex\)
(ACMMM159)
recognise and use the inverse relationship of the functions \(y=e^x\) and \(y=\ln x\)
(ACMMM160)
establish and use the formula \(\frac d{dx}\left(\ln x\right)=\frac1x\)
(ACMMM161)
establish and use the formula \(\int\frac1xdx=\ln\;x\;+c\) for \(x>0\)
(ACMMM162)
use logarithmic functions and their derivatives to solve practical problems.
(ACMMM163)