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9.055 Graphs of logarithmic functions

Worksheet
Graphs of logarithmic functions
1

Consider the function y = \log_{4} x and its given graph:

a

Complete the following table of values:

x\dfrac{1}{16}\dfrac{1}{4}416256
y
b

Find the x-intercept.

c

How many y-intercepts does the function have?

d

Find the x-value for which \log_{4} x = 1.

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2

Consider the functions graphed below:

Which of these graphs represents a logarithmic function of the form y = \log_{a} \left(x\right)?

A
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B
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C
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D
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3

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

a

Complete the following table of values:

x\dfrac{1}{2}12416
y
b

Sketch a graph of the function.

c

State the equation of the vertical asymptote.

4

Sketch the graph of y = \log_{5} x.

5

Consider the function y = \log_{4} x.

a

Complete the table of values.

x\dfrac{1}{1024}\dfrac{1}{4}1416256
y
b

Is \log_{4} x an increasing or decreasing function?

c

Describe the behaviour of \log_{4} x as x approaches 0.

d

State the value of y when x = 0.

6

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

a

For which of the following values of x will \log_{a} x be negative?

A

x = - 9

B

x = \dfrac{1}{9}

C

x = 9

D

\log_{a} x is never negative

b

For which of the following values of x will \log_{a} x be positive?

A

x = 5

B

x = - 5

C

x = \dfrac{1}{5}

D

\log_{a} x will never 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?

7

Consider the given graph of the logarithmic function y = \log_{a} x:

a

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

b

Which is a possible value for a,\dfrac{2}{3} or \dfrac{3}{2} ?

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8

Consider the functions y = \log_{2} x and y = \log_{3} x.

a

Sketch the two functions on the same set of axes.

b

Describe how the size of the base relates to the steepness of the graph.

9

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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Transformations of logarithmic functions
10

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

a

Complete the table of values below:

x\dfrac{1}{2}1248
f\left(x\right)=\log_2 x
g\left(x\right)=\log_2 x + 2
b

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

c

Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).

d

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.

11

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

a

Complete the table of values below:

x-8-4-2-1-\dfrac{1}{2}
f\left(x\right)=\log_2 \left( - x \right)
g\left(x\right)=\log_2 \left( - x \right) - 3
b

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

c

Describe a transformation that can be used to obtain g \left(x\right) from f \left(x\right).

d

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.

12

Sketch the graph of the following functions:

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

y = \log_{3} x translated 4 units down.

c

y= \log_{2} x + 4.

13

The graph of y = \log_{6} x is transformed to create the graph of y = \log_{6} x + 4. Describe a tranformation that could achieve this.

14

For each of the following functions:

i

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

ii

Sketch the translated graph.

a

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

b

y = \log_{3} \left( - x \right) translated upwards by 2 units.

15

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

a
y = \log_{6} x - 7
b
y = \log_{6} x +2
c
y = 3\log_{6} x
d
y = \log_{6} \left(x - 2\right)
16

Given the graph of y = \log_{8} \left( - x \right), sketch the graph of y = 3 \log_{8} \left( - x \right).

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17

Given the graph of y = \log_{2} x, sketch the graph of the following functions:

a
y = \dfrac{1}{3} \log_{2} x
b
y = - \dfrac{1}{2} \log_{2} x
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18

Find the equation of the following functions, given it is of the stated form:

a

y = k \log_{2} x

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b

y = 4 \log_{b} x

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c

y = \log_{4} x + c

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19

The function graphed has an equation of the form y = k \log_{2} x + c and passes through points A\left(4,11\right) and B\left(8,15\right):

a

Use the given points to form two equations relating c and k.

b

Hence, find the values of c and k.

c

State the equation of the function.

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