iGCSE (2021 Edition)

# 6.02 Logarithmic laws

Worksheet
Logarithmic laws
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

Evaluate the following logarithmic expressions:

a

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

b

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

c

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

d

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

e

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

f

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

3

Rewrite the following expressions without any powers or surds:

a

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

b

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

c

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

d

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

e
\log_{6} y^{5}
f
\log_{6} \sqrt{w}
g

\log_{10} \sqrt{10}

h

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

i

\log_{10} \left(10^a\right)

j
\log_{6} \left(36\right)^{5}
k

\log_{5} \sqrt{x^3}

l

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

4

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

a

\log_{9} \left(u v\right)

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

\log_{2} \left(5x\right)

d
\log_{10} \left(\dfrac{2}{x}\right)
e

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

f

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

g
\log \left(\dfrac{p q}{r}\right)
h

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

i

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

j

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

k

\log \left(\left( 14 x\right)^{\frac{1}{3}}\right)

l
\log \left(\sqrt{\frac{c^{8}}{d}}\right)
5

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

6

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

7

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

8

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)

9

Simplify each of the following expressions:

a

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

b

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

c

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

d

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

e

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

f

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

g

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

h

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

10

If \log_{a} 3 = 1.16 and \log_{a} 2 = 0.73, find the value of \log_{a} \sqrt{54}.

11

Using the following rounded values, evaluate the expressions below correct to three decimal places:

• \log_{10} 7 = 0.845

• \log_{10} 2 = 0.301

• \log_{10} 3 = 0.477

• \log_{10} 25 = 1.398

• \log_{10} 5 = 0.699

a

\log_{10} 28

b

\log_{10} \left(\dfrac{5}{3}\right)

c

\log_{10} 49 + \log_{10} 27

d

\log_{10} 49 - \log_{10} 125

12

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

13

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

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)

14

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

15

Rewrite the following in terms of base 10 logarithms:

a

\log_{4} 16

b

\log_{3} 0.9

c

\log_{3} \sqrt{5}

16

Consider the following logarithmic expressions:

i

Rewrite the expression in terms of base 10 logarithms.

ii

Hence, evaluate each to two decimal places.

a
\log_{8} 21
b
\log_{4} \sqrt{5}
c
\log_{2} 3^{3}
d
\log_{\pi} 105
17

If p = \log_c 8 and q = \log_c 10, write the following in terms of p and/or q:

a
\log_c 80
b
\log_c \left(\dfrac{4}{5}\right)
c
\log_c 100
d
\log_c 64c