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6.02 Properties of logarithms

Worksheet
Properties of logarithms
1

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

a

\log_{10} 5 + \log_{10} 4

b

\log_{10} 18 - \log_{10} 3

c

\log_{10} 7 - \log_{10} 28

d

\log_{5} 11 + \log_{5} 2 + \log_{5} 9

e

\log_{10} 5 + \log_{10} 7 - \log_{10} 3

f

\log_{7} 12 - \left(\log_{7} 2 + \log_{7} 3\right)

g

3 \left(\log_{10} 9 + \log_{10} 2\right)

h

3 \left(\log_{10} 6 - \log_{10} 2\right)

i

2 \log_{5} 22 - 2 \log_{5} 11

j

5 \log_{10} 6 + 5 \log_{10} 3

k

3 + \log_{4} 7

l
\log_{3} \left(5x\right) + \log_{3} \left(2y\right)
2

Simplify each of the following expressions in exact form without using a calculator:

a

\log 4 + \log 9

b

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

c

\log_{10} 11 + \log_{10} 2 + \log_{10} 9

d

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

e

\dfrac{\log_{10} 4}{\log_{10} 2}

f

\dfrac{\log_{4} 125}{\log_{4} 5}

g

\dfrac{\log a^{8}}{\log a^{4}}

h

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

i

\dfrac{\log \left(\dfrac{1}{x^{4}}\right)}{\log x}

j

\log_{10} 10 + \dfrac{\log_{10} \left(15^{20}\right)}{\log_{10} \left(15^{5}\right)}

k

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

l

10^{\log w}

m

\log 10 x + \log 10 y

n

x^{ 4 \log_{x} 3 - 6 \log_{x} 2}

o

y = \log_{a} \left(\sqrt{x} + \sqrt{x - 1}\right) + \log_{a} \left(\sqrt{x} - \sqrt{x - 1}\right)

3

Given that a > 1, fully simplify the following expressions:

a

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

b

\log_{a} \left(a^{9}\right)

c

\log_{a} \left(\dfrac{1}{a^{2}}\right)

d

\log_{a} \left(\sqrt{a}\right)

e

\log_{a} \left(\dfrac{1}{\sqrt{a}}\right)

4

Write each of the following as a single logarithm or integer:

a

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

b

5 \log x + 3 \log y

c

8 \log x - \dfrac{1}{3} \log y

d

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

e

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

f

5 \log_{10} 8 - 3 \log_{10} 4

g

2 \log_{6} 3 + \dfrac{1}{3} \log_{6} 64

h

\log_{2} 36 - 2 \log_{2} 3

5

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

6

Express the following as products:

a

\log_{a} A^{ - 2 }

b

\log_{6} \sqrt{w}

c

\log_{p} q^ r

d

\log_{3} B^ \frac{1}{3}

7

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

a

\log_{9} u v

b
\log_{5} \left(\dfrac{9}{7}\right)
c

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

d
\log_{b} \left(x^{2}\right)
e

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

f

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

g

\log \left(\left( 5 x\right)^{ - 7 }\right)

h

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

i

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

j
\log \left(\dfrac{p q}{r}\right)
k

\log \left(\left( 3 x + 7\right)^{ - 1 }\right)

l

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

8

Rewrite the expression \log x^{2} + \log x^{3} in the form k \log x.

9

Show that \log_{2} 5 = \dfrac{1}{\log_{5} 2}.

10

Amy has written the following:

\log_{b} 64 = \log_{b} \left( 64 \times 1\right) = \log_{b} 64 + \log_{b} 1

Is Amy correct? Explain your answer.

11

Rewrite the following in terms of base 10 logarithms:

a

\log_{4} 16

b

\log_{3} 0.9

c

\log_{3} \sqrt{5}

d

\log_{a}B

12

Rewrite \log_{3} 20 in terms of base 4 logarithms.

13

Use the properties of logarithms to evaluate the following expressions:

a

\log_{2} 16

b

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

c

\log_{5} 0.2

d

\log_{4} 1

e

\log_{36} 6

f

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

g

\log_{10} 0.1

h

\log_{7} \sqrt[3]{7}

i

2^{\log_{2} 3}

j

\log_{2} \sqrt[4]{2}

k

\log_{3} 3

l

\log_{16} \sqrt{2}

m

\log_{2} \left(\sqrt[3]{\dfrac{1}{16}}\right)

n

\log_{5} 125^{\frac{5}{4}}

14

Use the properties of logarithms to evaluate the following expressions:

a

\log_{10} 10^{\frac{5}{4}}

b

\log_{10} \left(10^{\sqrt{5}}\right)

c

\log_{16} \sqrt{2}

d

\log_{10} 2 + \log_{10} 5

e

\log_{4} 8 + \log_{4} 2

f

\log_{6} 12 + \log_{6} 18

g

\log_{2} 72 - \log_{2} 9

h

\log_{2} 36 - 2 \log_{2} 3

i

\log_{6} 12 + \log_{6} 15 - \log_{6} 5

j

\log_{3} 2 - \log_{3} 18

k

2 \log_{6} 3 + \dfrac{1}{3} \log_{6} 64

15

Consider the following logarithmic expressions:

i

Rewrite the expression in terms of base 10 logarithms.

ii

Hence, evaluate each correct to two decimal places.

a
\log_{8} 21
b
\log_{3} 15
c
\log_{2} 0.35
d
\log_{4} \sqrt{5}
16

If \log_{10} 6 = 0.778, calculate \log_{10} \left(\dfrac{1}{216}\right), without using a calculator.

17

If \log_{a} 3 = 1.16 and \log_{a} 2 = 0.73, find the value of \log_{a} \sqrt{54}, without using a calculator.

18

If \log_{k} a = 1.64, find the value of \log_{k} k a^{4}.

19

Using the rounded values \log_{x} 3 = 0.62 and \log_{x} 4 = 0.78, find the value of each of the following expressions:

a

\log_{x} 9

b

\log_{x} \sqrt{3}

c

\log_{x} 4 x

d

\log_{x} \dfrac{1}{3}

e

\log_{x} 36

20

Given that \log_{b} x = 2.6 and \log_{b} y = 4.2, determine the value of the following:

a

\log_{b} x^{3}

b

\log_{b} \sqrt[3]{y}

c

\log_{b} \left( x^{2} \sqrt{y}\right)

d

\log_{b} \left(\dfrac{b}{x}\right)

21

If \log_{x} 4 = 3.42 and \log_{x} 12 = 6.13, determine the value of the following:

a
\log_{x} \left(\dfrac{48}{x}\right)
b
\log_{x}3
c
\log_{x} \dfrac{4}{3}
d

\log_{x} \left(\dfrac{1}{3 x}\right)

22

Prove the following properties of logarithms:

a

\log_{a} x^{n} = n \log_{a} x

b
\log_{a} \left(\dfrac{x}{y}\right) = \log_{a} x - \log_{a} y
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ACMMM152

establish and use the algebraic properties of logarithms

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