Barnett/Ziegler/Byleen
College Algebra, 6th Edition
Chapter Five
Exponential & Logarithmic Functions
Copyright © 1999 by the McGraw-Hill Companies, Inc.
Basic Properties of the Graph of
f(x) = bx, b > 0, b  1
1. All graphs pass through the point (0, 1). b0 = 1 for any permissible
base b.
2. All graphs are continuous, with no holes or jumps.
3. The x axis is a horizontal asymptote.
4. If b > 1, then bx increases as x increases.
5. If 0 < b < 1, then bx decreases as x increases.
6. The function f is one-to-one.
y
y
8
6
4
x
1 
y =   = 2 –x
2 
–2
2
0
y = bx
0< b<1
y =2x
2
y = bx
b>1
1
x
DOMAIN = (–, )
x
RANGE = (0, )
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m

1 m
1 + 

m
1
2
10
2.59374…
100
2.70481…
1,000
2.71692…
10,000
2.71814…
100,000
2.71827…
1,000,000
2.71828…
.
.
.
The Number e
e = 2.718 281 828 459
-2
-1
0
1
2
3
4
2 e π
.
.
.
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Exponential Function with Base e
y = e–x
y = ex
y
20
10
–5
–3
–1
1
3
5
x
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Exponential Growth and Decay
Description
Equation
Graph
Uses
y
Unlimited
growth
Short-term population
growth (people, bacteria, etc.);
growth of money at continuous
compound interest
y = cekt
c, k > 0
c
t
0
y
Exponential
decay
ce–kt
y=
c, k > 0
Radioactive decay: light
absorption in water, glass, etc.;
atmospheric pressure;
electric circuits
c
0
t
5-2-52-1
Exponential Growth and Decay
Description
Equation
Graph
Uses
y
c
Limited
growth
y = c(1 – e–kt )
c, k > 0
Learning skills; sales fads;
company growth; electric circuits
t
0
y
Logistic
growth
M
1 + ce–kt
c, k, M > 0
M
Long-term population growth;
epidemics; sales of new products;
company growth
y=
0
t
5-2-52-2
Logarithmic Function with Base 2
f
y
y=2x
y= x
10
f
f –1
x =2y
or
y = log 2 x
5
x
–5
5
10
x
–3
–2
–1
0
1
2
3
y = 2x
x = 2y
1
8
1
4
1
2
1
8
1
4
1
2
1
2
4
8
1
2
4
8
f –1
y
–3
–2
–1
0
1
2
3
–5
DOMAIN of f = (– , ) = RANGE of f –1
RANGE of f = (0, ) = DOMAIN of
f –1
Ordered pairs
reversed
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Properties of Logarithmic Functions
1. logb 1 = 0
If b, M, and N are positive
real numbers, b  1,
and p and x
are real numbers, then:
2. logb b = 1
3. logb bx = x
4. b
logbx
= x, x > 0
5. logb MN = logb M + logb N
M
6. logb N = logb M – logb N
7. logb M p = p logb M
8. logb M = logb N
if and only if
M=N
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I
D = 10 log I
0
Decibel scale
Sound Intensity
Examples
Sound Intensity, W/m2
Sound
1.0  10–12
Threshold of hearing
5.2  10–10
Whisper
3.2  10–6
Normal conversation
8.5  10–4
Heavy traffic
3.2  10–3
Jackhammer
1.0  100
Threshold of pain
8.3  102
Jet plane with afterburner
5-4-55
2
E
M = 3 log E
0
Richter scale
Earthquake Intensity
Examples
Magnitude
on Richter Scale
M < 4.5
Destructive
Power
Small
4.5 < M < 5.5
Moderate
5.5 < M < 6.5
Large
6.5 < M < 7.5
Major
7.5 < M
Greatest
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Change-of-Base Formula
loga N
logb N = log b
a
Derivation:
logb
N= y
N=by
loga
N = loga b y
= y loga b
y
loga N
= log b
a
5-5-57