8.2 Transformations of Logarithmic Functions
Exponent
Exponent
Exponential Form: by = x.
Logarithmic form: y = logb x
Base
Base
y=x
f (x) = 2x
6
5
4
3
f (x) = log2 x
2
-2
-1
-1
-2
2
3 4
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6
1
y  a log c  b( x  h   k
Parameter Description of Transformation
Vertical stretch by a factor of |a|
a
1
|b|
Logarithmic
Transformations
Mapping Notation
(x, y) → (x, ay)
(x, y) →  x , y 


b
Horizontal stretch by a factor of
h
Horizontal translation right, h > 0
left, h < 0
(x, y) → (x + h, y)
k
Vertical translation up, k > 0
down, k < 0
(x, y) → (x, y + k)
|b|

Key Points from basic graph to transform include:
x-intercept (1, 0)
vertical asymptote x = 0
Reflections???
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Sketching Graph of Logarithms
y  log x
Do : x x  0, x 
Ra : y y 
y  a log c  b( x  h   k
y  log5 x
y  log(x  2)
Do : x x  0, x 
Ra : y y 
Do : x x  2, x 
Ra : y y 
Vertical asymptote
x=0
Vertical asymptote
x=0
Vertical asymptote
x=2
x-intercept at 1
x-intercept at 1
x-intercept at 3
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Determine the value of the missing coordinate.
The point (32, b) is on the graph of y  log2 x b = 5
b  log 2 32
The point (a, 3) is on the graph of
y  log 3 x a = 27
3  log 3 x
The point (a, 4) is on the graph of y  log2 x  1 a = 15
4  log 2 x 16, 4 
1

y


2
log
x

4
  1
The point (19, b) is on the graph of
5 
3

b = -3
1

y  2 log 5  19  4    1
3

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Describe the transformations on the graph of y = log2x
to become
y  log2 5(x  2)  3
Horizontal stretch by a factor of 1/5
Horizontal translation of 2 units left
Vertical translation of 3 units up
 9 
1, 0     , 3
5
The Vertical Asymptote , x = 0 translates to
The domain of the image graph is
x  2
 x | x  2, x 
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The graph of y = log x is transformed into the graph of
y + 3 = log(x - 6) by a translation of 6 units __i__ and 3 units _ii_.
The statement above is completed by the information in row
Row
i
ii
A
right
up
B
left
up
C
right
down
D
left
down
For the graph of y  log b (4 x  20) , where 0 < b < 1, the domain is
y  log b  4  x  5  
 x | x  5, x 
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The red graph can be generated by
stretching the blue graph of
y = log4 x. Write the equation that
describes the red graph.
y = log4 4x
The red graph can be generated by
stretching and reflecting the graph
of y = log4 x. Write the equation
that describes the red graph.
y = -3log4 x
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The solid graph can be generated
by translating the dashed graph of
y = log4 x. Write the equation that
describes the solid graph.
y = log4 (x) + 1
The graph of y = log2 x has been vertically stretched about the x-axis
by a factor of 3 , horizontally stretched about the y-axis by a factor of
1/5 , reflected in the x-axis, and translated of 7 units left and 2 units
up.
Write the equation of the transformed function in the form
y = a log2 (b(x - h) + k.
y = -3 log2 (5(x + 7) + 2
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Textbook p. 389 – 391
Level 1: (Basic Drill and Practice)
1 – 5, 7, 9
Level 2: (Problem Solving)
6, 8, 10, 11, 12, 13, 14
Level 3: (Extension and Higher Level)
15, 16, 17, C1, C2, C3
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