# 5.02 The exponential function

Worksheet
Exponential equations
1

Consider the following equations:

i

Rewrite each side of the equation with a base of 2.

ii

Hence, solve for x.

a

8^{x} = 4

b

16^{x} = \dfrac{1}{2}

c

\dfrac{1}{1024} = 4^x

d

\left(\sqrt{2}\right)^{x} = \sqrt[5]{32}

2

Solve the following exponential equations:

a

4^{x} = 4^{8}

b

3^{x} = 3^{\frac{2}{9}}

c

3^{x} = 27

d

7^{x} = 1

e

8^{x} = \dfrac{1}{8^{2}}

f

3^{y} = \dfrac{1}{27}

g

10^{x} = 0.01

h

3^{x} = 3^{6}

i

6^{x} = 6^{ - 3 }

j

6^{x} = 6^{\frac{4}{3}}

k

2^{x} = 64

l

9^{x} = 1

m

5^{x} = \dfrac{1}{5^{2}}

n

10^{x} = 0.0001

o

9^{y} = 81

3

Solve for x in the following equations:

a

9^{y} = 27

b

25^{y} = 125

c

3^{ 5 x - 10} = 1

d

25^{x + 1} = 125^{ 3 x - 4}

e

9^{x + 4} = 27^{x}

f

3^{ 4 x - 8} = 1

g

8^{x + 3} = 32^{ 2 x - 1}

h

30 \times 2^{x - 6} = 15

i

2^{x} \times 2^{x + 3} = 32

j

3^{x} \times 9^{x - k} = 27

k
a^{x-1} = a^4
l

a^{x + 1} = a^{3} \sqrt{a}

m

3^{x^{2} - 3 x} = 81

n

27 \left(2^{x}\right) = 6^{x}

o
3^{x} \times 3^{ n x} = 81
p

24 \times 2^{x - 6} = 12

q

2^{x} \times 2^{x + 2} = 16

r

3^{x} \times 9^{x - k} = 9

4

Solve the following exponential equations:

a

5^{x} = \sqrt[3]{5}

b

30^{n} = \sqrt[3]{30}

c

5^{x} = \sqrt{5}

d

5^{x} = \sqrt[4]{5}

e

9^{x} = \sqrt[9]{9}

f

\left(\sqrt{7}\right)^{y} = 49

g

\left(\sqrt{6}\right)^{y} = 36

h
\left(\sqrt{2}\right)^{k} = 0.5
i
20^{n} = \sqrt[4]{20}
5

Solve the following exponential equations:

a

\left(\dfrac{5}{8}\right)^{x} = \left(\dfrac{5}{8}\right)^{9}

b

7 \left(4^{x}\right) = \dfrac{7}{4^{3}}

c

\dfrac{1}{3^{x - 3}} = \sqrt[3]{9}

d

\left(\dfrac{1}{9}\right)^{x + 5} = 81

e

\left(\dfrac{1}{8}\right)^{x - 3} = 16^{ 4 x - 3}

f

\dfrac{25^{y}}{5^{4 - y}} = \sqrt{125}

g

8^{x + 5} = \dfrac{1}{32 \sqrt{2}}

h

\left(\dfrac{1}{25}\right)^{x - 4} = 125^{ 3 x - 1}

i

8^{x + 4} = \dfrac{1}{32 \sqrt{2}}

j

\left(\dfrac{1}{27}\right)^{x - 5} = 81

k

\dfrac{1}{5^{x + 3}} = \sqrt[3]{25}

l

9 \left(8^{x}\right) = \dfrac{9}{8^{2}}