Chapter 16

Exponential functions

16.1 Introduction to exponential functions

An exponential function is one which containsex,e
being a constant called the exponent and having an
approximate value of 2.7183. The exponent arises from
the natural laws of growth and decay and is used as a
base for natural or Napierian logarithms.
The most common method of evaluating an exponential
functionisby usingascientificnotationcalculator.Use
your calculator to check the following values.

e^1 = 2. 7182818 ,correct to 8 significant figures,

e−^1.^618 = 0. 1982949 ,correct to 7 significant figures,

e^0.^12 = 1. 1275 ,correct to 5 significant figures,

e−^1.^47 = 0. 22993 ,correct to 5 decimal places,

e−^0.^431 = 0. 6499 ,correct to 4 decimal places,

e^9.^32 = 11159 ,correct to 5 significant figures,

e−^2.^785 = 0. 0617291 ,correct to 7 decimal places.

Problem 1. Evaluate the following correct to 4

decimal places, using a calculator:

0. 0256 (e^5.^21 −e^2.^49 )

0. 0256 (e^5.^21 −e^2.^49 )

= 0. 0256 ( 183. 094058 ...− 12. 0612761 ...)

=4.3784,correct to 4 decimal places.

Problem 2. Evaluate the following correct to 4

decimal places, using a calculator:

5

(

e^0.^25 −e−^0.^25

e^0.^25 +e−^0.^25

)

5

(

e^0.^25 −e−^0.^25

e^0.^25 +e−^0.^25

)

= 5

(

1. 28402541 ...− 0. 77880078 ...

1. 28402541 ...+ 0. 77880078 ...

)

= 5

(

0. 5052246 ...

2. 0628262 ...

)

=1.2246,correct to 4 decimal places.

Problem 3. The instantaneous voltagevin a

capacitive circuit is related to timetby the equation

v=Ve−t/CRwhereV,CandRare constants.

Determinev, correct to 4 significant figures,

whent=50ms,C= 10 μF,R=47kand

V=300volts

v=Ve−t/CR= 300 e(−^50 ×^10

− (^3) )/( 10 × 10 − (^6) × 47 × 103 )
Using a calculator, v= 300 e−^0.^1063829 ...
= 300 ( 0. 89908025 ...)
=269.7volts.
Now try the following Practice Exercise
PracticeExercise 62 Evaluating
exponential functions (answers on page 347)

1. Evaluate the following,correct to 4 significant
   figures.
   (a) e−^1.^8 (b) e−^0.^78 (c) e^10


2. Evaluate the following,correct to 5 significant
   figures.
   (a) e^1.^629 (b) e−^2.^7483 (c) 0. 62 e^4.^178

DOI: 10.1016/B978-1-85617-697-2.00016-8