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Stage 5.1-3

8.05 Logarithmic graphs and scales

Worksheet
Graphs of logarithmic functions
1

For each of the following functions:

i

Complete the following table of values:

x0.3123451020
y
ii

Sketch the graph of the function on a number plane.

a
y = \log_{4} x
b
y = \log_{2} x
2

Consider the function y = \log_{3} x.

a

Complete the following table:

b

Sketch the graph of the function on a number plane.

PointCoordinates
A\left(\dfrac{1}{9},⬚\right)
B\left(\dfrac{1}{3}, ⬚\right)
C\left(1, ⬚\right)
D\left(3, ⬚\right)
E\left(9, ⬚\right)
F\left(⬚, 3\right)
G\left(⬚, 4\right)
H\left(⬚, 5\right)
3

Consider the function y = \log_{2} x.

a

Find the x-value of the x-intercept.

b

Complete the following table of values:

x\dfrac{1}{2}124
y
c

State the equation of the vertical asymptote.

d

Is \log_{2} x an increasing or decreasing function?

4

Consider the function y = \log_{\frac{1}{5}} x.

a

Find the x-value of the x-intercept.

b

Complete the table of values:

x\dfrac{1}{25}\dfrac{1}{5}1525
y
c

State the equation of the vertical asymptote.

d

Sketch the graph of the function on a number plane.

e

Is \log_{\frac{1}{5}} x an increasing or decreasing function?

5

Consider the function y = \log_{3} x.

a

Complete the following table of values:

x\dfrac{1}{3}\dfrac{2}{3}3981
y
b

Find the x-value of the x-intercept.

c

How many y-intercepts does the function have?

d

Find the x-value for which \log_{3} x = 1.

6

Sketch the graphs of y = \log_{3} x and y = \log_{5} x on the same set of axes.

7

Consider the given graph of f \left( x \right) = \log_{k} x:

a

Determine the value of the base k.

b

Hence, state the equation of f \left( x \right).

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8

Consider the function y = \log_{\frac{1}{3}} x.

a

Complete the following table of values:

x\dfrac{1}{81}\dfrac{1}{9}\dfrac{1}{3}13243
y
b

Is \log_{\frac{1}{3}} x an increasing or decreasing function?

c

Describe the behaviour of \log_{\frac{1}{3}} x as x approaches 0.

d

State the value of y when x = 0.

9

Consider the function y = \log_{a} x, where a is a value greater than 1.

a

For what values of x will \log_{a} x be negative?

b

For what values of x will \log_{a} x be positive?

c

Is there a value that \log_{a} x will always be greater than?

d

Is there a value that \log_{a} x will always be less than?

10

Consider the function y = \log_{a} x.

a

Determine whether the following values of a will make y = \log_{a} x an increasing function.

i

a = 0

ii

a = \dfrac{1}{2}

iii

a = 2

b

Determine whether the following values of a will make y = \log_{a} x a decreasing function.

i

a = 0

ii

a = \dfrac{1}{2}

iii

a = 2

c

Determine whether the following values of a will make y = \log_{a} x a relation, not a function.

i

a = e

ii

a = 1

iii

a = 0

d

When a = 1, what does the graph of the relation look like?

Transformations of logarithmic functions
11

Describe the following transformations:

a

The transformation of g \left( x \right) = \log_{3} x into f \left( x \right) = \log_{3} x + k, where k > 0.

b

The transformation of g \left( x \right) = \log_{3} x into f \left( x \right) = \log_{3} x + k, wherek < 0.

c

The transformation of g \left( x \right) = \log_{10} x into f \left( x \right) = a \log_{10} x, where a > 1.

d

The transformation of g \left( x \right) = \log_{2} x into f \left( x \right) = a \log_{2} x, where 0 < a < 1.

e

The transformation of g \left( x \right) = a \log_{10} x into f \left( x \right) = - a \log_{10} x.

12

Consider the functions f\left(x\right) = \log_{3} x and g\left(x\right) = \log_{3} x + 3:

a

Complete the table of values below:

x\dfrac{1}{3}139
f\left(x\right)=\log_3 x
g\left(x\right)=\log_3 x + 3
b

Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.

c

Determine whether each of the following features of the graph will remain unchanged after the given transformation:

i

The vertical asymptote.

ii

The general shape of the graph.

iii

The x-intercept.

iv

The range.

13

Consider the functions f\left(x\right) = \log_{2} \left( - x \right) and g\left(x\right) = \log_{2} \left( - x \right) - 2.

a

Complete the table of values below:

x-4-2-1-\dfrac{1}{2}
f\left(x\right)=\log_2 \left( - x \right)
g\left(x\right)=\log_2 \left( - x \right) - 2
b

Sketch the graphs of y = f\left(x\right) and y = g\left(x\right) on the same set of axes.

c

Determine whether each of the following features of the graph will remain unchanged after the given transformation:

i

The vertical asymptote.

ii

The general shape of the graph.

iii

The x-intercept.

iv

The domain.

14

The graph of y = \log_{6} x is transformed to create the graph of y = \log_{6} x -3. Describe a tranformation that could achieve this.

15

Consider the graph of y = \log_{6} x which has a vertical asymptote at x = 0. This graph is transformed to give the graph of y = \log_{6} x - 8. State the equation of the asymptote of the new graph.

16

Sketch the graph of the following functions:

a
y = \log_{3} x translated 2 units up.
b

y = \log_{2} x translated 4 units down.

c
f \left( x \right) = \log_{2} x + 5
d
f \left( x \right) = \log_{2} \left( - x \right) - 4
17

For each of the following functions:

i

State the equation of the function after it has been translated.

ii

Sketch the translated graph on a number plane.

a

y = \log_{5} x translated downwards by 2 units.

b

y = \log_{3} \left( - x \right) translated upwards by 3 units.

18

Consider the functions f \left( x \right) = 3 \log_{2} x and g \left( x \right) = \log_{2} x.

a

Evaluate g \left( 2 \right).

b

Evaluate f \left( 2 \right).

c

How does the graph of f \left( x \right) differ from the graph of g \left( x \right)?

19

Consider the graph of y = \log_{4} x which has a vertical asymptote at x = 0, state the equation of the asymptote of y = 4 \log_{4} x.

20

Consider the graphs of the functions f \left( x \right) = \log_{5} x and g \left( x \right):

Determine the equation of the function g \left( x \right).

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21

Consider the graphs of the functions f \left( x \right) = \log_{2} x and g \left( x \right):

Determine the equation of the function g \left( x \right).

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22

Consider the graph of y = \log_{9} x:

a

Describe the transformation of the graph of y = \log_{9} x to get the graph of y = - 2 \log_{9} x.

b

Hence, sketch the graph of \\y = - 2 \log_{9} x on the same set of axes as y = \log_{9} x.

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23

Sketch the graphs of the following functions on the same set of axes:

a

y = \log_{6} \left( - x \right) and y = 5 \log_{6} \left( - x \right)

b

y = \log_{2} x and y = \dfrac{1}{3} \log_{2} x

c

y = \log_{2} x and y = - \dfrac{1}{3} \log_{2} x

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Outcomes

MA5.3-11NA

uses the definition of a logarithm to establish and apply the laws of logarithms

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