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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[3]{6}

g

\sqrt[3]{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[3]{5}

i

30^{n} = \sqrt[3]{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[5]{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[3]{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

Which step was incorrect? Explain your answer.

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.

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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.

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