3.8
Solving equations involving
logarithms and exponentials
Introduction
It is often necessary to solve an equation in which the unknown occurs as a power, or exponent.
For example, you may need to find the value of x which satisfies 2x = 32. Very often the base
will be the exponential constant e, as in the equation ex = 20. To understand what follows
you must be familiar with the exponential constant. See leaflet 3.4 The exponential constant if
necessary.
You will also come across equations involving logarithms. For example you may need to find
the value of x which satisfies log10 x = 34. You will need to understand what is meant by a
logarithm, and the laws of logarithms (leaflets 2.19 What is a logarithm ? and 2.20 The laws of
logarithms). On this leaflet we explain how such equations can be solved.
1. Revision of logarithms
Logarithms provide an alternative way of writing expressions involving powers. If
a = bc then logb a = c
For example: 100 = 102 can be written as log10 100 = 2.
Similarly, e3 = 20.086 can be written as loge 20.086 = 3.
The third law of logarithms states that, for logarithms of any base,
log An = n log A
For example, we can write log10 52 as 2 log10 5, and loge 73 as 3 loge 7.
2. Solving equations involving powers
Example
Solve the equation ex = 14.
Solution
Writing ex = 14 in its alternative form using logarithms we obtain x = loge 14, which can be
evaluated directly using a calculator to give 2.639.
www.mathcentre.ac.uk 3.8.1 c Pearson Education Ltd 2000
Example
Solve the equation e3x = 14.
Solution
Writing e3x = 14 in its alternative form using logarithms we obtain 3x = loge 14 = 2.639. Hence
2.639
x= = 0.880.
3
To solve an equation of the form 2x = 32 it is necessary to take the logarithm of both sides of
the equation. This is referred to as ‘taking logs’. Usually we use logarithms to base 10 or base
e because values of these logarithms can be obtained using a scientific calculator.
Starting with 2x = 32, then taking logs produces log10 2x = log10 32. Using the third law of
logarithms, we can rewrite the left-hand side to give x log10 2 = log10 32. Dividing both sides by
log10 2 gives
log10 32
x=
log10 2
The right-hand side can now be evaluated using a calculator in order to find x:
log10 32 1.5051
x= = =5
log10 2 0.3010
Hence 25 = 32. Note that this answer can be checked by substitution into the original equation.
3. Solving equations involving logarithms
Example
Solve the equation log10 x = 0.98
Solution
Rewriting the equation in its alternative form using powers gives 100.98 = x. A calculator can
be used to evaluate 100.98 to give x = 9.550.
Example
Solve the equation loge 5x = 1.7
Solution
Rewriting the equation in its alternative form using powers gives e1.7 = 5x. A calculator can be
used to evaluate e1.7 to give 5x = 5.4739 so that x = 1.095 to 3dp.
Exercises
1. Solve each of the following equations to find x.
a) 3x = 15, b) ex = 15, c) 32x = 9, d) e5x−1 = 17, e) 103x = 4.
2. Solve the equations a) loge 2x = 1.36, b) log10 5x = 2, c) log10 (5x + 3) = 1.2.
Answers
1. a) 2.465, b) 2.708, c) 1, d) 0.767, e) 0.201.
2. a) 1.948, (3dp). b) 20, c) 2.570 (3dp).
www.mathcentre.ac.uk 3.8.2 c Pearson Education Ltd 2000
