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10&10a

5.04 Logarithms

Worksheet
Logarithms
1

Evaluate the following logarithmic expressions:

a

\log_{10} 1

b

\log_{10} 10

c

\log_{10} 100

d

\log_{10} 10\,000

e

\log_{10} 0.1

f

\log_{10} 0.001

g

3\log_{10} {100}

h

\dfrac{\log_{10} {0.01}}{\log_{10} {1000}}

2

Evaluate \log_{10} 45, to two decimal places.

3

Evaluate \log_{10} 5.16, correct to three decimal places.

4

Evaluate the following logarithmic expressions:

a

\log_{2} 8

b

\log_{4} 16

c

\log_{5} 1

d

\log_{5} 125

e

\log_{2} 16

f

\log_{3} 3

g

\log_{5} 0.2

h

\log_{4} 1

i

\log_{10} \left(\dfrac{1}{10}\right)

j

\log_{8} \left(\dfrac{1}{64}\right)

k

\log_{2} \left(\dfrac{1}{4}\right)

l

\log_{2} \left(\dfrac{1}{8}\right)

5

Evaluate the following logarithmic expressions:

a

\log_{0.2} 25

b

\log_{\frac{1}{10}} 10\,000

c

\log_{2} 0.25

d

\log_{0.9} 0.81

6

Evaluate the following logarithmic expressions:

a

\log_{36} 6

b

\log_{25} 5

c

\log_{8} 2

d

\log_{27} 3

e

\log_{10} {\sqrt{10}}

f

\log_{10} \dfrac{1}{\sqrt{10}}

g

\log_{3}{\sqrt{3}}

h

\log_{5}{\left(\sqrt[3]{5}\right)}

Logarithmic and exponential form
7

Rewrite each of the following equations in logarithmic form:

a

5^{2} = 25

b

3^{x} = 81

c

3^{1} = 3

d

2^{0} = 1

e

4.4^{0} = 1

f

4^{\frac{5}{2}} = 32

g

4^{ - 2 } = 0.0625

h

4^{ - 3 } = \dfrac{1}{64}

i

x^{1.5} = 64

j

25^{\frac{3}{2}} = 125

k

4^{ - 1 } = 0.25

l

2^{ - 6 } = \dfrac{1}{64}

m

4^{2} = 16

n

4^{1} = 4

o

3^{0} = 1

p

\left(9.8\right)^{0} = 1

8

Rewrite each of the following equations in logarithmic form:

a

10^{2} = 100

b

10^{ - 2 } = \dfrac{1}{100}

c

10^{4} = 10\,000

d

10^{ - 3 } = \dfrac{1}{1000}

e

10^{\frac{1}{2}} = \sqrt{10}

f

10^{ - \frac{1}{2} } = \dfrac{1}{\sqrt{10}}

g

10^{-1} = 0.1

h

10^{-4} = 0.000\,1

9

Rewrite each of the following equations in logarithmic form:

a

x^{2.5} = 243

b

9^{x} = 81

10

Rewrite each of the following equations in exponential form:

a

\log_{10} 1000 = 3

b

\log_{10} 100\,000 = 5

c

\log_{10} 0.1 = - 1

d

\log_{10} \left(\sqrt{10}\right) = \dfrac{1}{2}

e

\log_{10} \left(\dfrac{1}{1000}\right) = - 3

f

\log_{10} \left(\dfrac{1}{\sqrt{10}}\right) = -\dfrac{1}{2}

11

Rewrite each of the following equations in exponential form:

a

\log_{4} 16 = 2

b

\log_{5} 5 = 1

c

\log_{2} 0.125 = - 3

d

\log_{3} \dfrac{1}{3} = - 1

e

\log_{5.8} 33.64 = 2

f

\log_{\frac{1}{3}} 9 = - 2

g

\log_{6} 36 = 2

h

\log_{2} 2 = 1

i

\log_{8} 1 = 0

j

\log_{7} \dfrac{1}{7} = - 1

k

\log_{8.4} 70.56 = 2

l

\log_{\frac{1}{3}} 81 = - 4

12

Rewrite each of the following equations in exponential form:

a

\log_{x} 64 = 2

b

\log_{5} x = 9

c

\log_{x} 32= 5

d

\log_{8} x = 6

13

For each of the following equations:

i

Rewrite the equation in logarithmic form.

ii

Approximate the value of x to two decimal places.

a
10^{x} = 860
b
10^{x} = 0.0002
c
10^{x} = 790
d
10^{x} = 0.0003
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Outcomes

ACMNA265 (10a)

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

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