Topics
20. Logarithmic and Exponential Functions
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iGCSE (2021 Edition)

20.04 Exponential equations

Lesson
Worksheet
Practice
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 equations:

a
4^{x} = 4^{8}
b
9^{x} = 9^{ - 4 }
c
3^{x} = 3^{\frac{2}{9}}
d
3^{x} = 27
e
7^{x} = 1
f
10^{x} = 0.01
g
\left(\dfrac{8}{3}\right)^{x} = \left(\dfrac{8}{3}\right)^{7}
h
8^{x} = \dfrac{1}{8^{2}}
i
7 \left(4^{x}\right) = \dfrac{7}{4^{3}}
j
5^{x} = \sqrt[3]{5}
k
8^{x} = \sqrt{8}
l
30^{n} = \sqrt[3]{30}
3

Find the interval, of two consecutive integers, 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
4

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}

5

Solve the following exponential equations:

a

9^{y} = 27

b
9^{x + 3} = 27^{x}
c

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

d

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

e

3^{ 5 x - 10} = 1

f
5^{ - 3 x -1} = 3125
g
5^{ 10 x + 33} = 125
h

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

i

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

j

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

k

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

l

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

m

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

n

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

o
\left(2^{2}\right)^{x + 7} = 2^{3}
p
\left(2^{4}\right)^{ 2 x - 10} = 2^{2}
q

81^{x - 1} = 9^{ 3 x + 5}

r

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

s
a^{x-1} = a^4
t

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

u

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

v

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

6

For each of the following equations:

i

Simplify the left side of the equation.

ii

Solve the equation for x.

a
3^{x} \times 3^{ n x} = 81
b
3^{x} \times 9^{x - k} = 27
7

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
8

Given that 2^{x} \times 5^{x + 2} = 50, find the value of 2^{x + 2} \times 5^{x}.

Applications
9

Find the x-coordinate of the point of intersection of the graphs of y = 2^{ 5 x} and y = 4^{x - 3}.

10

Find the value of h, given the point \left(h,\dfrac{1}{9}\right) lies on the curve y = 3^{ - x }.

11

Given the points \left(3, n\right), \left(k, 16\right) and \left(m, \dfrac{1}{4}\right) all lie on the curve with equation y = 2^{x}, find the value of:

a

n

b

k

c

m

12

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:

a
32
b

1024

c

4096

13

The frequency f \left(\text{Hz}\right) of the nth key of an 88-key piano is given by f \left( n \right) = 440 \left(2^{\frac{1}{12}}\right)^{n - 49}.

a

Find the frequency of the forty-ninth key.

b

Find the frequency of the 40th key to the nearest whole number.

c

Find the value of n that corresponds to the key with a frequency of 1760 \text{ Hz}.

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Outcomes

0607E3.10A

Logarithmic function as the inverse of the exponential function y = a^x equivalent to x = log_a y. Solution to a^x = b as x = log b/log a.

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