Chapter 16
Exponential functions
16.1 Introduction to exponential
functions
An exponential function is one which contains e x , 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
function is by using a scientific notation calculator. 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 voltage v in a
capacitive circuit is related to time t by the equation
v = V e
−t /CR where V , C and R are constants.
Determine v, correct to 4 significant figures,
when t = 50 ms, C = 10 μF, R = 47 k and
V = 300 volts
v = V e
−t /CR
= 300e
(−50×10 −3 )/(10×10 −6 ×47×10 3 )
Using a calculator,
v = 300e
−0.1063829...
= 300(0.89908025 ...)
= 269.7 volts.
Now try the following Practice Exercise
Practice Exercise 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) 0.62e 4.178
DOI: 10.1016/B978-1-85617-697-2.00016-8
Exponential functions
16.1 Introduction to exponential
functions
An exponential function is one which contains e x , 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
function is by using a scientific notation calculator. 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 voltage v in a
capacitive circuit is related to time t by the equation
v = V e
−t /CR where V , C and R are constants.
Determine v, correct to 4 significant figures,
when t = 50 ms, C = 10 μF, R = 47 k and
V = 300 volts
v = V e
−t /CR
= 300e
(−50×10 −3 )/(10×10 −6 ×47×10 3 )
Using a calculator,
v = 300e
−0.1063829...
= 300(0.89908025 ...)
= 269.7 volts.
Now try the following Practice Exercise
Practice Exercise 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) 0.62e 4.178
DOI: 10.1016/B978-1-85617-697-2.00016-8