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VCE 12 Methods 2023

4.09 The natural logarithm

Worksheet
The natural logarithm
1

Describe the logarithmic function known as \ln x in terms of its base.

2

Write the following equations in logarithmic form.

a
e^{0} = 1
b
e^{x} = 3
c
e^{\frac{1}{2}} = \sqrt{e}
d
e^{p} = q
3

Rewrite the following logarithmic equations using index notation:

a

\ln e = 1

b

\ln \sqrt{e} = \dfrac{1}{2}

c

\ln \left(\dfrac{1}{\sqrt{e}}\right) = - \dfrac{1}{2}

d

\ln 5 = x

4

Simplify the following:

a
\log_{e} e
b
\log_{e} 1
c
\ln_{e} e^3
d
e^{\ln w}
5

Is the value of \log_{e} 2 greater than or less than 1?

6

Consider the following logarithmic expressions:

\log_{e} 7, \log_{3} 7, \log_{2} 7

Which expression has the largest value? Explain your answer.

7

Evaluate each of the following expressions:

a

\ln e^{3.5}

b

\ln e^{4}

c

\sqrt{6} \ln \left(e^{\sqrt{6}}\right)

d

\ln \left(\dfrac{1}{e^{2}}\right)

8

Rewrite each of the following expressions as a single logarithm:

a

\ln 3 + \ln 5

b

\ln 24 - \ln 4

c

\ln 8 - \ln 32

d

\ln 11 + \ln 5 + \ln 8

e

\ln 24 - \left(\ln 2 + \ln 6\right)

f

3 \ln \left(x^{5}\right) - 4 \ln \left(x^{2}\right)

g

6 \ln x - \dfrac{1}{6} \ln y

h

7 x + \ln \left(1 + e^{ - 7 x }\right)

9

Simplify each of the following expressions:

a

\ln 7 + \ln 15

b

\ln e + \ln e

c

3 \left(\ln 7 + \ln 3\right)

d

2 \left(\ln 4 - \ln 2\right)

e

\dfrac{\ln 49}{\ln 7}

f

5 \ln 16 - 5 \ln 8

g

4 \ln 3 - 12 \ln 9

h

\dfrac{\ln \left(\dfrac{1}{x^{2}}\right)}{\ln x}

i

\dfrac{\ln a^{6}}{\ln a^{3}}

j

\ln e + \dfrac{\ln \left(e^{42}\right)}{\ln \left(e^{7}\right)}

k

\dfrac{48 \ln \sqrt{e}}{\ln \left(e^{6}\right)}

l

\dfrac{\ln \left(a^{4}\right)}{\ln \sqrt[3]{a}}

10

Rewrite each of the following as a sum or difference of logarithms:

a

\ln \left( u v\right)

b

\ln \left( 3 x\right)

c

\ln \left( 2 p^2 \right)

d

\ln \left( x^2 y^7 \right)

e

\ln \left(\dfrac{19}{7}\right)

f
\ln \left(\dfrac{15}{8}\right)
g
\ln \left(\dfrac{1}{x}\right)
h

\ln \left(\sqrt{\dfrac{c^{8}}{d}}\right)

11

Use the properties of logarithms to express each of the following without any powers or surds:

a
\ln \left(x^{2}\right) + \ln \left(x^{3}\right)
b
\ln \left(\dfrac{1}{y^{2}}\right)
12

Use the properties of logarithms to rewrite each of the following in the form \ln b^{a} where \\a > 0:

a
- 2 \ln x
b
\dfrac{\ln x}{2}
13

Michael has written \ln 25 = \ln \left( 25 \times 1\right) = \ln 25 + \ln 1. Is Michael correct? Explain your answer.

14

Suppose u = \ln a and v = \ln b. Rewrite the following expressions in terms of u and v:

a

\ln \left(ab\right)

b

\ln \left( b^{8} \sqrt{a}\right)

c

\ln \left(\dfrac{a^{5}}{b^{4}}\right)

d

\ln \left(\sqrt{\dfrac{a^{7}}{b^{3}}}\right)

15

Consider x=\ln 31. Find the value of x, correct to two decimal places.

16

Find the value of each of the following correct to four decimal places:

a

\ln 94

b

\ln 0.042

c

\ln \left( 18 \times 35\right)

d

\ln \left( 10 - \sqrt 144 \right)

17

Find the exact value of x in each of the following:

a

3 \ln x = 9

b

\ln 3 x = 5

c

2 \ln 3 x = 6

d

5 e^{x} = 25

e

4 e^{x + 7} = 20

f

\ln \left( 2 x - 1\right) = \ln \left(x + 3\right)

g

\ln x + 2 = \ln \left( 2 x + 1\right)

h

e^{ - 5 \ln x } = \dfrac{1}{243}

i

\ln e^{x} - 3 \ln e = \ln e^{2}

j

e^{x + \ln 8} = 5 e^{x} + 3

k

\ln e^{\ln \left(x - 1\right)} - \ln \left(x - 7\right) = \ln 4

l

e^{ 2 x} - 8 e^{x} + 7 = 0

m

e^{ 2 x} - 7 e^{x} + 12 = 0

n

3 e^{ 2 x} - 2 e^{x} = 8

o

\dfrac{1}{3} e^{ 2 x} + 2 e^{x} = 9

p

\ln^{2} x - 4 \ln x = 5

18

Rewrite y = 2 \log_{e} x - 3 with x as the subject of the equation.

19

Find the value of y in each of the following expressions:

a

\ln \left(\ln e^{ - y }\right) = \ln 3

b

e^{ - \ln y } = 3

Graphs of natural logarithm functions
20

Consider the function f \left( x \right) = \ln x.

a

Complete the following table of values, correct to two decimal places:

x\dfrac{1}{4}\dfrac{1}{3}\dfrac{1}{2}1234
f \left( x \right)
b

Hence, sketch the graph of f \left( x \right) = \ln x.

c

State the equation of the vertical asymptote of f \left( x \right) = \ln x.

d

State the equation of the x-intercept of f \left( x \right) = \ln x.

21

Consider the graph of the functions \\y = \log_{2} x and y = \log_{3} x:

Use the approximation e = 2.718.

a

For what values of x will the graph of \\ y = \log_{e} x lie above the graph of \\ y = \log_{3} x and below the graph of \\ y = \log_{2} x?

b

For what values of x will the graph of \\ y = \log_{e} x lie above the graph of \\ y = \log_{2} x and below the graph of \\ y = \log_{3} x?

-1
1
2
3
4
5
6
x
-4
-3
-2
-1
1
2
3
4
y
22

Describe the transformation of the following:

a

g \left( x \right) = \ln x into f \left( x \right) = \ln x + k, where k > 0.

b

g \left( x \right) = \ln x into f \left( x \right) = \ln x + k, where k < 0.

c

g \left( x \right) = \ln x into f \left( x \right) = \ln \left(x - h\right), for h > 0.

d

g \left( x \right) = \ln x into f \left( x \right) = \ln \left(x - h\right), for h < 0.

23

Consider the following translations:

i

State the direction in which the graph has been translated.

ii

State the equation of the vertical asymptote of the function.

a

The graph of y = \ln x is translated to create the graph of y = \ln x + 4.

b

The graph of y = \ln x is translated to create the graph of y = \ln \left(x + 8\right).

24

Consider the following graphs of f \left( x \right) = \ln x and g \left( x \right) below:

a

Describe the transformation required to get from f \left( x \right) to g \left( x \right).

b

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

a
-1
1
2
3
4
5
6
7
8
x
-4
-3
-2
-1
1
2
3
4
5
6
7
y
b
-1
1
2
3
4
5
6
7
x
-4
-3
-2
-1
1
2
3
4
y
c
-1
1
2
3
4
5
6
7
8
x
-8
-6
-4
-2
2
4
6
8
y
d
-1
1
2
3
4
5
6
7
8
x
-8
-6
-4
-2
2
4
6
8
y
25

The given graph of f \left( x \right) is the result of two transformations of y = \ln x:

a

Describe the transformations required for y = \ln x to become f \left( x \right).

b

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

1
2
3
4
5
6
7
8
x
-2
-1
1
2
3
4
y
26

The given graph f \left( x \right) is the result of two transformations of y = \ln x:

a

Describe the transformations required for y = \ln x to become f \left( x \right).

b

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

1
2
3
4
5
x
-4
-3
-2
-1
1
2
3
4
5
y
27

Consider the function f \left( x \right) = e^{x}.

a

Sketch the graph of the function, the line y = x and the inverse function f^{-1} \left( x \right) on the same set of axes.

b

State the kind of function the inverse of f \left( x \right) = e^{x} is.

c

Hence, state the equation of the inverse function.

28

For each of the following functions:

i

Rewrite f \left( x \right) as a sum in simplified form.

ii

Describe how the graph of f \left( x \right) can be obtained from the graph of y = \ln x.

a

f \left( x \right) = \ln \left( e^{2} x\right)

b

f \left( x \right) = \ln \left(\dfrac{x}{e^{3}}\right)

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Outcomes

U34.AoS1.2

graphs of the following functions: power functions, y=x^n; exponential functions, y=a^x, in particular y = e^x ; logarithmic functions, y = log_e(x) and y=log_10(x) ; and circular functions, 𝑦 = sin(𝑥) , 𝑦 = cos (𝑥) and 𝑦 = tan(𝑥) and their key features

U34.AoS1.7

the key features and properties of a function or relation and its graph and of families of functions and relations and their graphs

U34.AoS1.18

sketch by hand graphs of polynomial functions up to degree 4; simple power functions, y=x^n where n in N, y=a^x, (using key points (-1, 1/a), (0,1), and (1,a); log x base e; log x base 10; and simple transformations of these

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