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AustraliaNSW
Stage 5.1-3

8.04 Logarithm laws

Worksheet
Logarithm laws
1

Rewrite the following as the sum or difference of logarithms without any powers or surds:

a

\log_{8} x y

b
\log_{3} \left(\dfrac{7}{11}\right)
c
\log \left(\dfrac{p q}{r}\right)
d

\log \left(\dfrac{1}{x y}\right)

e

\log \left(\left( 7 x\right)^{6}\right)

f

\log \left( 7 x^{4}\right)

g

\log \left(\left( 2 x\right)^{ - 1 }\right)

h

\log \left(\dfrac{7}{2}\right)

i

\log \left( 5 x^{ - 1 }\right)

j

\log \left(\left( 3 x\right)^{ - 7 }\right)

k

\log \left( 7 x^{ - 6 }\right)

l

\log \left( 5 x^{\frac{2}{3}}\right)

m

\log \left(\left( 14 x\right)^{\frac{2}{3}}\right)

n

\log \left(\sqrt{\frac{c^{4}}{d}}\right)

o

\log \left(v^{2}\right)

2

Rewrite the following expressions without any powers or surds:

a

\log_{4} y^{7}

b

\log_{8} \left(x^{6}\right)

c

\log \left(x^{4}\right)

d

\log \left(\left(x + 7\right)^{6}\right)

e

\log \left(\left( 2 x + 5\right)^{6}\right)

f

\log_{a} B^{ - 5 }

g

\log \left(\left( 2 x + 3\right)^{ - 1 }\right)

h

\log \left(\left( 4 x + 3\right)^{ - 8 }\right)

i

\log \left(x^{\frac{1}{5}}\right)

j

\log_{5} \sqrt{y}

k

\log \left(\left( 5 x + 9\right)^{\frac{1}{3}}\right)

3

Write each of the following expressions as a single logarithmic term:

a

\log 7 + \log 12

b

\log_{10} 11 + \log_{10} 5 + \log_{10} 3

c

\log_{10} 12 - \left(\log_{10} 2 + \log_{10} 3\right)

d

\log_{10} 42 - \log_{10} 7

e

\log_{10} 8 - \log_{10} 32

f

\log_{10} 2 + \log_{10} 3 - \log_{10} 7

g

5 \left(\log_{10} 3 + \log_{10} 6\right)

h

3 \left(\log_{10} 6 - \log_{10} 3\right)

i

3 \log_{10} 22 - 3 \log_{10} 11

j

7 \log_{10} 5 - 21 \log_{10} 25

4

Write each of the following expressions as a single logarithmic term:

a

3 \log x^{5} - 2 \log x^{4}

b

6 \log x + 5 \log y

c

8 \log x - \dfrac{1}{2} \log y

d

5 \log x - \log \left(\dfrac{1}{x}\right) - \log y

e

\log x^{4} + \log x^{2}

f

\log 2 x + \log 50 y

5

Rewrite each of the following equations in index notation:

a

\log_{10} 1000 = 3

b

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

c

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

d

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

6

Rewrite each of the following equations in logarithmic notation:

a

10^{4} = 10\,000

b

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

c

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

d

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

7

Simplify each of the following expressions:

a

\log_{10} 10 + \dfrac{\log_{10} \left(12^{8}\right)}{\log_{10} \left(12^{2}\right)}

b

\dfrac{8 \log_{10} \left(\sqrt{10}\right)}{\log_{10} \left(100\right)}

c

\log_{10} \left(10\right) + \log_{10} \left(10\right)

d

\dfrac{\log_{10} 125}{\log_{10} 5}

e

\dfrac{\log_{4} 49}{\log_{4} 7}

8

Simplify each of the following expressions:

a

\dfrac{5 \log m^{2}}{6 \log \sqrt[3]{m}}

b

\dfrac{\log a^{4}}{\log a^{2}}

c

\dfrac{\log a^{3}}{\log \sqrt[3]{a}}

d

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

9

Write \log \left(\dfrac{2 u}{3 v}\right) in terms of \log 2, \log u, \log 3 and \log v.

10

Rewrite the following in terms of \log u and \log v without any powers or surds:

a

\log \left( u^{3} v^{5}\right)

b

\log \left(\dfrac{\sqrt[3]{v}}{\sqrt{u}}\right)

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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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