iGCSE (2021 Edition)

6.01 Definition of the logarithm

Worksheet
Logarithms
1

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

25^{1.5} = 125

g

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

h

4^{ - 2 } = 0.0625

i

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

j

x^{1.5} = 64

2

Rewrite each of the following equations in logarithmic form:

a

10^{2} = 100

b

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

c

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

d

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

3

Rewrite each of the following equations in exponential form:

a

\log_{4} 16 = 2

b

\log_{5} 5 = 1

c

\log_{8} 1 = 0

d

\log_{2} 0.125 = - 3

e

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

f

\log_{5.8} 33.64 = 2

g

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

h

\log_{x} 32= 5

i

\log_{8} x = 6

4

Rewrite each of the following equations in exponential form:

a

\log_{10} 10\,000 = 4

b

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

c

\log_{10} \left(\dfrac{1}{10\,000}\right) = - 4

d

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

5

True or false: The equation 4 \left(1.09\right)^{x} = 20 is equivalent to x = \log_{1.09} 5.

6

The logarithm of what number is equal to zero?

7

Evaluate the following logarithmic 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}}

8

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

9

In order to estimate \log_{10}90 without a calculator:

a

Evaluate \log_{10}10.

b

Evaluate \log_{10}100.

c

Between which two consecutive integers would you expect \log_{10}90 to lie?

10

Consider the expression \log 8892.

a

Between which two consecutive integers would you expect \log 8892 to lie?

b

Find the value of \log 8892 correct to four decimal places.

11

Consider the function f \left( x \right) = \log_{10} \left( - 6 x \right). Determine whether each of the following is defined. If so, evaluate the expression correct to two decimal places.

a

f \left( 3 \right)

b

f \left( \dfrac{1}{3} \right)

c

f \left( - \dfrac{1}{3} \right)

d

f \left( - 3 \right)

12

Consider the function f \left( x \right) = \log_{2} \left(x - 2\right). Evaluate f \left( 10 \right).

13

Consider the function f \left( x \right) = \log_{2}x + 3. Evaluate:

a

f \left( 8 \right)

b

f \left( \dfrac{1}{8} \right)

c

f \left( 32 \right)

d

f \left( \dfrac{1}{64} \right)

14

True or false: If 5^{x - 6} = 2^{ 4 x + 3}, then x - 6 = \log_{5} 2^{ 4 x + 3}.

15

Consider the function f \left( x \right) = \log_{10} \left(4 x\right). Solve for the value of x for which f \left( x \right) = 2.

16

Consider the function f \left( x \right) = \log_{10} \left( 3 x + 9\right). Evaluate f \left( 4 \right) to two decimal places.

17

Consider the function f \left( x \right) = \log_{4} \left(k x\right) + 1. Solve for the value of k for which f \left( 4 \right) = 3.

18

Consider the function f \left( x \right) = 4 \log_{\frac{1}{4}} x.

a

Evaluate f \left( \dfrac{1}{4} \right).

b

Solve the value for x for which f \left( x \right) = 0.

19

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} = 790
b
10^{x} = 0.0003
20

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)