ENGINEERING MATHEMATICS

2. TRANSCENDENTAL FUNCTIONS

BACKGROUND


The functions that involve a combination of basic arithmetic operations, powers, or roots are
called algebraic functions. Most of the functions studied so far are algebraic functions. The
set of transcendental functions includes the trigonometric, inverse trigonometric, exponential,
and logarithmic functions. In this chapter, we turn to exponential and logarithmic functions.
These functions are used to describe phenomena ranging from growth of investments to
decay of radioactive materials, which cannot be described with algebraic functions. Since the
exponential and logarithmic functions transcend what can be described with algebraic
functions, they are called transcendental 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 exponential and logarithmic functions.




COMPILED BY T. PAEPAE

,TABLE OF CONTENTS


2. TRANSCENDENTAL FUNCTIONS ............................................................................... 0


2.1 EXPONENTS AND THE EXPONENTIAL FUNCTION ............................................ 2

2.1.1 Defining Exponential Functions ........................................................................ 3

2.1.2 Exponential Laws and Properties ..................................................................... 4

2.1.3 Simplifying and Evaluating Exponential Expressions ....................................... 6

2.1.4 The Solution of Exponential Equations........................................................... 12

2.1.5 The Exponential Function with Base e (Euler’s Number)............................... 18


2.2 LOGARITHMIC RULES AND EQUATIONS ......................................................... 22

2.2.2 Defining Logarithmic Functions ...................................................................... 23

2.2.3 Converting Between Logarithmic Form and Exponential Form ....................... 23

2.2.4 Laws and Properties of Logarithms ................................................................ 24

2.2.5 Common and Natural Logarithms .................................................................. 33

2.2.6 Using a Calculator to Evaluate Logarithms .................................................... 35

2.2.7 Logarithmic Equations ................................................................................... 36

2.2.8 Exponential Equations Using Logarithms ....................................................... 39


2.3 MANIPULATION OF EQUATIONS BY CHANGING THE SUBJECT ................... 45

2.3.2 Manipulation of a Formula.............................................................................. 46

2.3.3 Manipulation of Formulae Containing Logarithms and Exponents .................. 48


2.4 GRAPHS OF THE EXPONENTIAL AND LOGARITHMIC FUNCTIONS .............. 52

2.4.2 Definition and Graphical Representation of the Exponential Function ............ 53

2.4.3 The Graph of the Logarithmic Function .......................................................... 61




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,2.1 EXPONENTS AND THE EXPONENTIAL FUNCTION



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.




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, INTRODUCTION


Many of the functions we’ve studied so far included exponents. But in every case, the
exponent was a constant, and the base was often a variable as shown below:


Index or exponent or power

4a 3

coefficient base


The use of powers (also called exponents) provides a convenient form of algebraic
shorthand. Repeated factors of the same base, for example a  a  a can be written as a 3 ,
where the number 3 indicates the number of factors multiplied together. In general, the
product of n such factors a , where a and n are positive integers, is written a n , where a is

called the base and n is called the index or exponent or power. Any number multiplying a n
is called the coefficient.


2.1.1 Defining Exponential Functions

Let’s start by noting that f and g given by

f ( x)  2 x and g ( x)  x 2

are not the same function. The function f ( x )  2 x is called an exponential function because
the variable, x , is the exponent. It should not be confused with the power function

g ( x )  x 2 , in which the variable is the base. The exponential function f will be discussed in
this section. The domain of this function is the set of real numbers.


In general, the exponential functions are functions of the form
f ( x)  a x
where a  0 , a  1 , and x is any real number. The base a  1 is excluded because it yields

f ( x)  1x  1 , which is a constant function, not an exponential function.


Exponential functions are useful for modeling many natural phenomena, such as population
growth (if a  1 ) and radioactive decay (if a  1 ).




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