# Topic 9: Functions and logarithms ```HEATHCOTE HIGH SCHOOL
YEAR 10 MATHEMATICS PROGRAM
2006
TOPIC:
Functions and Logarithms
OUTCOMES:
Stage 5.3: PAS5.3.8 Describes, interprets and sketches functions and uses the definition of a logarithm to establish
and apply the laws of logarithms (p 109)
SUGGESTED TIME:
CONTENT
Key Ideas for Stage 5.3
1. Define functions
2. Use function notation
3. Determine inverse functions
4. Establish and apply the laws of logarithms
KNOWLEDGE AND SKILLS
 defining a function as a rule or relationship
where for each input there is only one output,
or that associates every member of one set
with exactly one member of the second set
 using the vertical line test on a graph to
decide whether it represents a function
 using the notation f(x)
 using f(c) notation to determine the value of
f(x) when x  c
 finding the permissible x - and y -values for a
variety of functions
 determining the inverse functions for a
variety of functions and recognising their
graphs as reflections in the line y = x
 sketching the graph of y  f ( x )  k and
y  f ( x  a ) given the graph of
y  f (x )
 The logarithm of a number to any positive
base is the index when the number is
expressed as a power of the base.
Alternatively,
a x  y  log a y  x
where
a  0, y  0
.
 defining logarithms as indices and translating
index statements into equivalent statements
using logarithms eg
Heathcote High School
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RESOURCES
17 p464 - 488
New Century 10 Adv &amp; Int BLM
14.5
Excel 10 Advanced p87 - 95
WORKING
MATHEMATICALLY
TERMINOLOGY
interpret, functions, logarithm,
relationship, vertical line test,
permissible, parabola, hyperbola,
exponential, inverse, reflection,
base, index, power, function
exponents, contrast, compare
DIAGNOSIS/ASSESSMENT
 sketch graphs to model
relationships that occur in
practical situations and
explain the relationship
between the variables
represented in the graph
(Applying Strategies,
Reasoning, Communicating)
 explain whether a given graph
represents a function
(Applying Strategies,
Reasoning, Communicating)
 recognise and describe the
restrictions that need to be
so that they have an inverse
function (Applying Strategies,
Reasoning, Reflecting,
Communicating
 relate logarithms to practical
scales that use indices
eg Richter, decibel and pH
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9  32 ,  log 3 9  2
1
1
 2 1 ,  log 2  1
2
2
3
4 2  8,  log 4 8 
3
2

 deducing the following laws of logarithms
from the laws of indices:
(Applying Strategies,
Communicating)
 compare and contrast a set of
exponential and logarithmic
graphs drawn on the same
axes to determine similarities
and differences
eg
y  2x
y  log 2 x
log a x  log a y  log a ( xy )
x
log a x  log a y  log a ( )
y
y  3x
y  log 3 x
n
 log a x  n log a x

 establishing and using the following results:
(Applying Strategies,
Reasoning, Communicating)
log a a x  x
log a a  1
log a 1  0
1
log a ( )   log a x
x
 applying the laws of logarithms to evaluate
simple expressions
eg evaluate the following:
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
 simplifying expressions using the laws of
logarithms
5 log a  log a 4
a
a
eg simplify
 drawing the graphs of the inverse functions
y  a x and y  log a x
 solving simple equations that contain
exponents or logarithms
2t  8, 4t 1 
eg
Heathcote High School
1
8 2
log 27 3  x, log 4 x  2
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Heathcote High School
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