EXPONENTIAL AND LOGARITHMIC
FUNCTIONS
INTRODUCTION
So far, we have dealt mainly with algebraic functions, which include polynomial functions
and rational functions. In this chapter, you will study two types of non-algebraic functions –
exponential functions and logarithmic functions. These functions are examples of
transcendental functions because they are said to “transcend” or go beyond algebra and
are used to describe phenomena such as the growth of a population, the growth of an
investment that earns compound interest, or the decay of a radioactive substance, which
cannot be described with algebraic functions.
LEARNING OUTCOMES
On completion of this chapter, you will be able to:
Apply the exponential laws and properties to evaluate, manipulate, and simplify
exponential expressions containing exponents.
Solve exponential equations.
Perform calculations with Euler’s number.
Evaluate, manipulate, and simplify logarithmic expressions.
Solve logarithmic equations.
Manipulate and change the subject of formulae containing logarithms and exponents.
Sketch the graphs of exponential and logarithmic functions.
COMPILED BY T. PAEPAE
,2.1 EXPONENTIAL FUNCTIONS
Why it is important to understand: Exponential Functions
“Exponential functions are used in engineering, physics, biology and economics. There are
many quantities that grow exponentially; some examples are population, compound interest
and charge in a capacitor. With exponential growth, the rate of growth increases as time
increases. We also have exponential decay; some examples are radioactive decay,
atmospheric pressure, Newton’s law of cooling and linear expansion. Understanding and
using exponential functions is important in many branches of engineering”. Bird, J., 2017.
Higher engineering mathematics. Routledge.
SPECIFIC OUTCOMES
On completion of this study unit, you will be able to:
Apply the exponential laws and properties to evaluate, manipulate, and simplify
exponential expressions containing exponents.
Use factorisation and exponents to simplify expressions.
Use calculators to do evaluations.
Solve exponential equations.
Perform calculations with Euler’s number.
1
,INTRODUCTION
In this section, we study a new class of functions called exponential functions. For example,
𝑓𝑓(𝑥𝑥) = 2𝑥𝑥
is an exponential function (with base 2). Notice how quickly the values of this function
increase.
𝑓𝑓(3) = 23 = 8
𝑓𝑓(10) = 210 = 1024
𝑓𝑓(30) = 230 = 1073741824
Compare this with the function 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 2 , where 𝑔𝑔(𝑥𝑥) = 302 = 900. The point is that when
the variable is in the exponent, even a small change in the variable can cause a dramatic
change in the value of the function.
2.1.1 Defining Exponential Functions
The equation
𝑓𝑓(𝑥𝑥) = 𝑎𝑎 𝑥𝑥 where 𝑎𝑎 > 0, 𝑎𝑎 ≠ 1
defines an exponential function for each different constant 𝑎𝑎, called the base. The
independent variable 𝑥𝑥 can assume any real value.
2.1.2 Evaluating Exponential Functions
Recall that the base of an exponential function must be a positive real number other than 1.
Why do we limit the base 𝑎𝑎 to positive values? To ensure that the outputs will be real
numbers. Observe what happens if the base is not positive:
1
1 1
Let 𝑎𝑎 = −9 and 𝑥𝑥 = . Then 𝑓𝑓 � � = (−9)2 = √−9, which is not a real number.
2 2
Why do we limit the base to positive values other than 1? Because base 1 results in the
constant function. Observe what happens if the base is 1:
Let 𝑎𝑎 = 1. Then 𝑓𝑓(𝑥𝑥) = 1𝑥𝑥 = 1 for any value of 𝑥𝑥.
When evaluating exponential functions with a calculator, remember to enclose fractional
exponents in parentheses. Because the calculator follows the order of operations,
parentheses are crucial in order to obtain the correct result.
2
, 2.1.3 Exponential Laws and Properties
Exponential functions whose domains include irrational numbers obey the familiar laws of
exponents for rational exponents. We summarize these exponent laws and properties here.
These laws and properties are used to simplify expressions.
Multiplying expressions involving exponents
𝑎𝑎𝑚𝑚 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚+𝑛𝑛
Dividing expressions involving exponents
𝑎𝑎 𝑚𝑚
𝑎𝑎 𝑛𝑛
= 𝑎𝑎𝑚𝑚−𝑛𝑛
Multiple indices
(𝑎𝑎𝑚𝑚 )𝑛𝑛 = 𝑎𝑎𝑚𝑚𝑚𝑚
These three basic laws lead to a number of important results or properties
𝑎𝑎 𝑚𝑚
𝑎𝑎0 = 1 because 𝑎𝑎𝑚𝑚 ÷ 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚−𝑛𝑛 and also 𝑎𝑎𝑚𝑚 ÷ 𝑎𝑎𝑛𝑛 =
𝑎𝑎 𝑛𝑛
𝑎𝑎𝑚𝑚
Then if 𝑛𝑛 = 𝑚𝑚, 𝑎𝑎𝑚𝑚−𝑚𝑚 = 𝑎𝑎0 and = 1. So 𝑎𝑎0 = 1
𝑎𝑎𝑚𝑚
1 𝑎𝑎−𝑚𝑚 ×𝑎𝑎𝑚𝑚 𝑎𝑎0 1 1
𝑎𝑎−𝑚𝑚 = because 𝑎𝑎 −𝑚𝑚 = = 𝑎𝑎𝑚𝑚 = 𝑎𝑎𝑚𝑚. So 𝑎𝑎−𝑚𝑚 = 𝑎𝑎𝑚𝑚
𝑎𝑎 𝑚𝑚 𝑎𝑎𝑚𝑚
1
With similar reasoning, = 𝑎𝑎𝑚𝑚
𝑎𝑎 −𝑚𝑚
1 1 𝑚𝑚 𝑚𝑚 1
𝑚𝑚 𝑚𝑚
𝑎𝑎𝑚𝑚 = √𝑎𝑎 because �𝑎𝑎 𝑚𝑚 � = 𝑎𝑎𝑚𝑚 = 𝑎𝑎1 = 𝑎𝑎. So 𝑎𝑎𝑚𝑚 = √𝑎𝑎
𝑛𝑛 𝑛𝑛
𝑚𝑚 𝑚𝑚
From this, it follows that 𝑎𝑎𝑚𝑚 = √𝑎𝑎𝑛𝑛 or � √𝑎𝑎�
(𝑎𝑎𝑎𝑎)𝑚𝑚 = 𝑎𝑎𝑚𝑚 𝑏𝑏𝑚𝑚 e.g. (2𝑥𝑥 𝑛𝑛 )4 = 24 𝑥𝑥 4𝑛𝑛 = 16𝑥𝑥 4𝑛𝑛
𝑎𝑎 𝑚𝑚 𝑎𝑎 𝑚𝑚 2 3 23 8
�𝑏𝑏 � = 𝑏𝑏𝑚𝑚 e.g. �𝑥𝑥� = 𝑥𝑥 3 = 𝑥𝑥 3
𝑎𝑎 −𝑚𝑚 𝑏𝑏 𝑚𝑚 𝑏𝑏𝑚𝑚 3 −2 2 2 22 4
�𝑏𝑏 � =� � = 𝑎𝑎𝑚𝑚 e.g. � 2� = � � = 32 =
𝑎𝑎 3 9
3
FUNCTIONS
INTRODUCTION
So far, we have dealt mainly with algebraic functions, which include polynomial functions
and rational functions. In this chapter, you will study two types of non-algebraic functions –
exponential functions and logarithmic functions. These functions are examples of
transcendental functions because they are said to “transcend” or go beyond algebra and
are used to describe phenomena such as the growth of a population, the growth of an
investment that earns compound interest, or the decay of a radioactive substance, which
cannot be described with algebraic functions.
LEARNING OUTCOMES
On completion of this chapter, you will be able to:
Apply the exponential laws and properties to evaluate, manipulate, and simplify
exponential expressions containing exponents.
Solve exponential equations.
Perform calculations with Euler’s number.
Evaluate, manipulate, and simplify logarithmic expressions.
Solve logarithmic equations.
Manipulate and change the subject of formulae containing logarithms and exponents.
Sketch the graphs of exponential and logarithmic functions.
COMPILED BY T. PAEPAE
,2.1 EXPONENTIAL FUNCTIONS
Why it is important to understand: Exponential Functions
“Exponential functions are used in engineering, physics, biology and economics. There are
many quantities that grow exponentially; some examples are population, compound interest
and charge in a capacitor. With exponential growth, the rate of growth increases as time
increases. We also have exponential decay; some examples are radioactive decay,
atmospheric pressure, Newton’s law of cooling and linear expansion. Understanding and
using exponential functions is important in many branches of engineering”. Bird, J., 2017.
Higher engineering mathematics. Routledge.
SPECIFIC OUTCOMES
On completion of this study unit, you will be able to:
Apply the exponential laws and properties to evaluate, manipulate, and simplify
exponential expressions containing exponents.
Use factorisation and exponents to simplify expressions.
Use calculators to do evaluations.
Solve exponential equations.
Perform calculations with Euler’s number.
1
,INTRODUCTION
In this section, we study a new class of functions called exponential functions. For example,
𝑓𝑓(𝑥𝑥) = 2𝑥𝑥
is an exponential function (with base 2). Notice how quickly the values of this function
increase.
𝑓𝑓(3) = 23 = 8
𝑓𝑓(10) = 210 = 1024
𝑓𝑓(30) = 230 = 1073741824
Compare this with the function 𝑔𝑔(𝑥𝑥) = 𝑥𝑥 2 , where 𝑔𝑔(𝑥𝑥) = 302 = 900. The point is that when
the variable is in the exponent, even a small change in the variable can cause a dramatic
change in the value of the function.
2.1.1 Defining Exponential Functions
The equation
𝑓𝑓(𝑥𝑥) = 𝑎𝑎 𝑥𝑥 where 𝑎𝑎 > 0, 𝑎𝑎 ≠ 1
defines an exponential function for each different constant 𝑎𝑎, called the base. The
independent variable 𝑥𝑥 can assume any real value.
2.1.2 Evaluating Exponential Functions
Recall that the base of an exponential function must be a positive real number other than 1.
Why do we limit the base 𝑎𝑎 to positive values? To ensure that the outputs will be real
numbers. Observe what happens if the base is not positive:
1
1 1
Let 𝑎𝑎 = −9 and 𝑥𝑥 = . Then 𝑓𝑓 � � = (−9)2 = √−9, which is not a real number.
2 2
Why do we limit the base to positive values other than 1? Because base 1 results in the
constant function. Observe what happens if the base is 1:
Let 𝑎𝑎 = 1. Then 𝑓𝑓(𝑥𝑥) = 1𝑥𝑥 = 1 for any value of 𝑥𝑥.
When evaluating exponential functions with a calculator, remember to enclose fractional
exponents in parentheses. Because the calculator follows the order of operations,
parentheses are crucial in order to obtain the correct result.
2
, 2.1.3 Exponential Laws and Properties
Exponential functions whose domains include irrational numbers obey the familiar laws of
exponents for rational exponents. We summarize these exponent laws and properties here.
These laws and properties are used to simplify expressions.
Multiplying expressions involving exponents
𝑎𝑎𝑚𝑚 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚+𝑛𝑛
Dividing expressions involving exponents
𝑎𝑎 𝑚𝑚
𝑎𝑎 𝑛𝑛
= 𝑎𝑎𝑚𝑚−𝑛𝑛
Multiple indices
(𝑎𝑎𝑚𝑚 )𝑛𝑛 = 𝑎𝑎𝑚𝑚𝑚𝑚
These three basic laws lead to a number of important results or properties
𝑎𝑎 𝑚𝑚
𝑎𝑎0 = 1 because 𝑎𝑎𝑚𝑚 ÷ 𝑎𝑎𝑛𝑛 = 𝑎𝑎𝑚𝑚−𝑛𝑛 and also 𝑎𝑎𝑚𝑚 ÷ 𝑎𝑎𝑛𝑛 =
𝑎𝑎 𝑛𝑛
𝑎𝑎𝑚𝑚
Then if 𝑛𝑛 = 𝑚𝑚, 𝑎𝑎𝑚𝑚−𝑚𝑚 = 𝑎𝑎0 and = 1. So 𝑎𝑎0 = 1
𝑎𝑎𝑚𝑚
1 𝑎𝑎−𝑚𝑚 ×𝑎𝑎𝑚𝑚 𝑎𝑎0 1 1
𝑎𝑎−𝑚𝑚 = because 𝑎𝑎 −𝑚𝑚 = = 𝑎𝑎𝑚𝑚 = 𝑎𝑎𝑚𝑚. So 𝑎𝑎−𝑚𝑚 = 𝑎𝑎𝑚𝑚
𝑎𝑎 𝑚𝑚 𝑎𝑎𝑚𝑚
1
With similar reasoning, = 𝑎𝑎𝑚𝑚
𝑎𝑎 −𝑚𝑚
1 1 𝑚𝑚 𝑚𝑚 1
𝑚𝑚 𝑚𝑚
𝑎𝑎𝑚𝑚 = √𝑎𝑎 because �𝑎𝑎 𝑚𝑚 � = 𝑎𝑎𝑚𝑚 = 𝑎𝑎1 = 𝑎𝑎. So 𝑎𝑎𝑚𝑚 = √𝑎𝑎
𝑛𝑛 𝑛𝑛
𝑚𝑚 𝑚𝑚
From this, it follows that 𝑎𝑎𝑚𝑚 = √𝑎𝑎𝑛𝑛 or � √𝑎𝑎�
(𝑎𝑎𝑎𝑎)𝑚𝑚 = 𝑎𝑎𝑚𝑚 𝑏𝑏𝑚𝑚 e.g. (2𝑥𝑥 𝑛𝑛 )4 = 24 𝑥𝑥 4𝑛𝑛 = 16𝑥𝑥 4𝑛𝑛
𝑎𝑎 𝑚𝑚 𝑎𝑎 𝑚𝑚 2 3 23 8
�𝑏𝑏 � = 𝑏𝑏𝑚𝑚 e.g. �𝑥𝑥� = 𝑥𝑥 3 = 𝑥𝑥 3
𝑎𝑎 −𝑚𝑚 𝑏𝑏 𝑚𝑚 𝑏𝑏𝑚𝑚 3 −2 2 2 22 4
�𝑏𝑏 � =� � = 𝑎𝑎𝑚𝑚 e.g. � 2� = � � = 32 =
𝑎𝑎 3 9
3
