EXPONENTIAL AND
LOGARITHMIC MODELS
3.5
COMMON MODELS
  Exponential

y = aebx
  Gaussian

y = ae-(x-b)^2/c
  Logistic

growth
y = a/(1+be-rx)
  Logarithmic

y = a + b(lnx)
4. Logistic
growthy model:
=
a
1. Exponential growth
model:
= ae”,
b > 0y 3.5
Section
and Logarithmic Models
1 + be~
1 + Exponential
be~
2. Exponential decay model:
y = ae~, b > 0
5. Logarithmic
models:
y = ay +
blnx,
y = ya =
+ ablogx
5. Logarithmic
models:
= a
+ blnx,
+ blogx
3. Gaussian model:
y =
The basic
shapes
of theofgraphs
of these
functions
are shown
in Figure
3.29.3.29.
The basic
shapes
the graphs
of these
functions
are shown
in Figure
4. Logistic growth model:
y =
a
1 + be~
4
4
4
Logarithmic
at5. you
should learnmodels:
y
a
=
+
blnx,
y
a + blogx
=
Introduction
cognize
the five
mostofcom
The basic
shapes
the graphs of these functions are shown in Figure 3.29.
4
4
257
—2
—2
EXPONENTIAL GROWTH MODEL
3-
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p
1
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—2
MODEL
NATURAL LOGARITHMI
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3EXPONENTIAL GROWTH MODEL
3-
2-
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—i —i
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EXPONENTIAL DECAY MODEL
3-
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i
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I
—2
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I I
2
2
2
3
3
on types of models involving
The five most common types of mathematical models involving exponential 2ponential and logarithmic
functions and logarithmic functions are as follows.
2
2
nctions.
4
1. Exponential growth model:2
y = ae”, b > 0
e exponential
growth and
cay functions
to model
x
2. Exponential
decay model: x y x = ae~, b > 0
x
3
I]
—i —i
—I
~
—3 —1
—2 —1
—I
I ~ I3. Gaussian
—3 —2
ve real-life proble’~
p
model: —1 —1
y =
2
—j
—j
—1
e Gaussian funct
—2
and solve re —2 —2
—2
4. Logistic growth model:
y =
a
1 + be~
oblen’
x
x
GAUSSIAN
MODELEXPONENTIAL
EXPONENTIAL
DECAY
GAUSSIAN
MODELMODEL y = a + blogx
~ logistic
gi I EXPONENTIAL
GROWTH
DECAY—i
MODELMODEL y
—I
~ EXPONENTIAL
—3GROWTH
—2
—1
5.MODEL
Logarithmic
models:
= a + blnx,
LOGI5TIC GROWTH
—j
—1
odel and
The basic shapes of the graphs of these functions are shown in Figure 3.29.
FIGURE 3.29
I
EXPONENTIAL DECAY
—j
—I
ix
I ~
(I
—2
~+lnx~
~+lnx~
2
GAUSSIAN MODEL
2
ix
ix
(I (I
—I
—2x
—1-
—1I
—3
I
—2
I
—2—2-
—1
I ~
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x
x
~
x x
7/—i
—1
—I
GROWTH
NATURAL
LOGARITHMIC
COMMON
LOGARITHMIC
p LOGI5TIC
—1- MODELMODEL
LOGI5TIC
GROWTH
MODELMODEL
NATURAL
LOGARITHMIC
COMMON
LOGARITHMIC
MODELMODEL
1
—2
—2
FIGURE
3.29
FIGURE 3.29
—2
—2GAUSSIAN MODEL
EXPONENTIAL GROWTH MODEL
EXPONENTIAL DECAY MODEL
can often
of insight
a situation
modeled
You You
can often
gain gain
quite quite
a bit aofbitinsight
into into
a situation
modeled
by anby an
exponential
or logarithmic
function
by
identifying
and MODEL
interpreting
the function’s
LOGI5TIC GROWTH
MODEL
LOGARITHMIC
MODEL
COMMON
LOGARITHMIC
exponential
or NATURAL
logarithmic
function
by identifying
and interpreting
the function’s
asymptotes.
Usegraphs
the graphs
in Figure
to identify
the asymptotes
the graph
FIGURE 3.29
asymptotes.
Use the
in Figure
3.29 3.29
to identify
the asymptotes
of theofgraph
~+lnx~
32
of each
function.
of each
function.
You can often gain quite a bit of insight into a situation modeled by an
2exponential or logarithmic function by identifying
and interpreting the function’s
asymptotes. Use the graphs in Figure 3.29 to identify the asymptotes of the graph
You can often gain quite a bit of insi
exponential or logarithmic function by ident
asymptotes. Use the graphs in Figure 3.29 to
of each function.