 iGCSE (2021 Edition)

6.04 Solving exponential equations

Worksheet
Power equations
1

Solve the following equations:

a

x^{5} = 3^{5}

b

8^{ - 7 } = x^{ - 7 }

c

x^{3} = \left(\dfrac{8}{5}\right)^{3}

d

x^{ - 7 } = \dfrac{1}{6^{7}}

e

3 \left(x^{ - 9 }\right) = \dfrac{3}{2^{9}}

f

x^{\frac{1}{3}} = \sqrt{6}

g

\sqrt{5} = x^{\frac{1}{3}}

h

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

Exponential equations
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

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

h

5^{x} = \sqrt{5}

i

30^{n} = \sqrt{30}

j

10^{x} = 0.01

3

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{32}

4

Solve the following exponential equations:

a

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

b

\left(\sqrt{2}\right)^{k} = 0.5

c

9^{y} = 27

d

3^{ 5 x - 10} = 1

e

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

f

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

g

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

h

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

i

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

j

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

k

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

l

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

m

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

n
a^{x-1} = a^4
o

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

p

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

q

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

r
3^{x} \times 3^{ n x} = 81
5

Consider the following equations:

i

Determine the substitution, m that would reduce the equation to a quadratic.

ii

Hence, solve the equation for x.

a
\left(2^{x}\right)^{2} - 9 \times 2^{x} + 8 = 0
b
2^{ 2 x} - 12 \times 2^{x} + 32 = 0
c
4 \times 2^{ 2 x} - 34 \times 2^{x} + 16 = 0
d
4^{ 2 x} - 65 \times 4^{x} + 64 = 0
e
4^{ x} - 5 \times 2^{x} + 4 = 0
f
9^{ x} - 12 \times 3^{x} + 27 = 0
Exponential equations and logarithms
6

Find the interval in which the solution of the following equations will lie:

a
3^{x} = 57
b
3^{x} = 29
c
2^{x} = \dfrac{1}{13}
d
2^{x} = - 5
7

Consider the following equations:

i

Rearrange the equation into the form x = \dfrac{\log A}{\log B}.

ii

Evaluate x to three decimal places.

a

13^{x} = 5

b

5^{x} = \dfrac{1}{11}

c

3^{x} = 2

d

4^{x} = 6.4

e

\left(0.4\right)^{x} = 5

f

5^{x} + 4 = 3129

g

2^{ - x } = 6

h

27^{x} + 4 = 19\,211

8

Consider the equation 4^{ 2 x - 8} = 70.

a

Make x the subject of the equation.

b

Evaluate x to three decimal places.

9

For each of the following incorrect sets of working:

i

ii

Rearrange the original equation into the form a = \dfrac{\log A}{\log B}.

iii

Evaluate a to three decimal places.

a

\begin{aligned} 9 ^ {a} &= 40 \\ \log9^{a} &= \log40 & (1)\\ a + \log 9 &= \log40 & (2)\\ a &= \log40 - \log9 & (3)\\ &\approx 0.648 & (4) \end{aligned}

b

\begin{aligned} 2 ^ {a} &= 89 \\ \log2^{a} &= \log89 & (1) \\ a \log 2 &= \log89 & (2)\\ a &= \log_{89} 2 & (3)\\ &\approx 0.154 & (4) \end{aligned}

Applications
10

A certain type of cell splits in two every hour and each cell produced also splits in two each hour. The total number of cells after t hours is given by:

N(t)=2^t

Find the time when the number of cells will reach the following amounts. Round your answers to two decimal places where necessary.

a
32
b

1024

c

3000

11

A population of mice, t months after initial observation, is modelled by:

P(t)=500(1.2^t)
a

State the initial population.

b

By what percentage is the poulation increasing by each month?

c

Find the time when the population reaches 1500 to two decimal places.

12

A population of wallabies, t years after initial observation, is modelled by:

P(t)=800(0.85^t)
a

State the initial population.

b

By what percentage is the poulation decreasing by each year?

c

Find the time when the population reaches 200 to two decimal places.

13

A microbe culture initially has a population of 900\,000 and the population increases by 40\% every hour. Let t be the number of hours passed.

Find the time when the population reaches 7\,200\,000 to three decimal places.

Outcomes

0606C7.2

Know and use the laws of logarithms (including change of base of logarithms).

0606C7.3

Solve equations of the form a^x = b.