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Exponential and Logarithmic Functions
Chapter 4
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Composite Functions
Section 4.1
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Composite Functions
Construct new function from two given functions f and g
Composite function:
Denoted by f ° g
Read as “f composed with g”Defined by
(f ° g)(x) = f(g(x))
Domain: The set of all numbers x in the domain of g such that g(x) is in the domain of f.
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Composite Functions
Note that we perform the inside function g(x) first.
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Composite Functions
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Composite Functions
Example. Suppose that f(x) = x3 — 2 and g(x) = 2x2 + 1. Find the values of the following expressions.
(a) Problem: (f ◦ g)(1)
Answer:
(b) Problem: (g ◦ f)(1)
Answer:
(c) Problem: (f ◦ f)(0)
Answer:
![Page 7: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/7.jpg)
Composite Functions
Example. Suppose that f(x) = 2x2 + 3 and g(x) = 4x3 + 1.
(a) Problem: Find f ◦ g.
Answer:
(b) Problem: Find the domain of f ◦ g.
Answer:
(c) Problem: Find g ◦ f.
Answer:
(d) Problem: Find the domain of f ◦ g.
Answer:
![Page 8: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/8.jpg)
Composite Functions
Example. Suppose that f(x) = and g(x) =
(a) Problem: Find f ◦ g.
Answer:
(b) Problem: Find the domain of f ◦ g.
Answer:
(c) Problem: Find g ◦ f.
Answer:
(d) Problem: Find the domain of f ◦ g.
Answer:
![Page 9: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/9.jpg)
Composite Functions
Example.
Problem: If f(x) = 4x + 2 and
g(x) = show that for all x,
(f ◦ g)(x) = (g ◦ f)(x) = x
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Decomposing Composite Functions
Example.
Problem: Find functions f and g such that
f ◦ g = H if
Answer:
![Page 11: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/11.jpg)
Key Points
Composite Functions
Decomposing Composite Functions
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One-to-One Functions;Inverse Functions
Section 4.2
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One-to-One Functions
One-to-one function: Any two different inputs in the domain correspond to two different outputs in the range.
If x1 and x2 are two different inputs of a
function f, then f(x1) ≠ f(x2).
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One-to-One Functions
One-to-one function
Not a one-to-one function
Not a function
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One-to-One Functions
Example.Problem: Is this function one-to-one?
Answer:
Melissa
John
Jennifer
Patrick
$45,000
$40,000
$50,000
Person Salary
![Page 16: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/16.jpg)
One-to-One Functions
Example.Problem: Is this function one-to-one?
Answer:
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935
Person ID Number
![Page 17: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/17.jpg)
One-to-One Functions
Example. Determine whether the following functions are one-to-one.
(a) Problem: f(x) = x2 + 2
Answer:
(b) Problem: g(x) = x3 — 5
Answer:
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One-to-One Functions
Theorem. A function that is increasing on an interval I is a one-to-one function on I.
A function that is decreasing on an interval I is a one-to-one function on I.
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Horizontal-line Test
If every horizontal line intersects the graph of a function f in at most one point, then f is one-to-one.
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Horizontal-line Test
Example.
Problem: Use the graph to determine whether the function is one-to-one.
Answer:
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
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Horizontal-line Test
Example.
Problem: Use the graph to determine whether the function is one-to-one.
Answer:
-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
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Inverse Functions
Requires f to be a one-to-one function
The inverse function of fWritten f—1
Defined as the function which takes f(x) as input
Returns the output x.
In other words, f—1 undoes the action of f
f—1(f(x)) = x for all x in the domain of f
f(f—1(x)) = x for all x in the domain of f—1
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Inverse Functions
Example. Find the inverse of the function shown.
Problem:
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935
Person ID Number
![Page 24: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/24.jpg)
Inverse Functions
Example. (cont.)
Answer:
Alex
Kim
Dana
Pat
1451678
1672969
2004783
1914935
PersonID Number
![Page 25: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/25.jpg)
Inverse Functions
Example.
Problem: Find the inverse of the function shown.
{(0, 0), (1, 1), (2, 4), (3, 9), (4, 16)}
Answer:
![Page 26: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/26.jpg)
Domain and Range of Inverse Functions
If f is one-to-one, its inverse is a function.
The domain of f—1 is the range of f.
The range of f—1 is the domain of f
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Domain and Range of Inverse Functions
Example.
Problem: Verify that the inverse of
f(x) = 3x — 1 is
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Graphs of Inverse Functions
The graph of a function f and its inverse f—1 are symmetric with respect to the line y = x.
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-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Graphs of Inverse Functions
Example.
Problem: Find the graph of the inverse function
Answer:
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Finding Inverse Functions
If y = f(x),
Inverse if given implicitly by x = f(y).
Solve for y if possible to get y = f —1(x)
Process
Step 1: Interchange x and y to obtain an equation x = f(y)
Step 2: If possible, solve for y in terms of x.
Step 3: Check the result.
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Finding Inverse Functions
Example.
Problem: Find the inverse of the function
Answer:
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Restricting the Domain
If a function is not one-to-one, we can often restrict its domain so that the new function is one-to-one.
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-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Restricting the Domain
Example.
Problem: Find the inverse of if the domain of f is x ≥ 0.
Answer:
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Key Points
One-to-One Functions
Horizontal-line Test
Inverse Functions
Domain and Range of Inverse Functions
Graphs of Inverse Functions
Finding Inverse Functions
Restricting the Domain
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Exponential Functions
Section 4.3
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Exponents
For negative exponents:
For fractional exponents:
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Exponents
Example.
Problem: Approximate 3π to five decimal places.
Answer:
![Page 38: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/38.jpg)
Laws of Exponents
Theorem. [Laws of Exponents]
If s, t, a and b are real numbers with a > 0
and b > 0, then
as · at = as+t
(as)t = ast
(ab)s = as · bs
1s = 1
a0 = 1
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Exponential Functions
Exponential function: function of the form
f(x) = ax
where a is a positive real number (a > 0)
a ≠ 1.
Domain of f: Set of all real numbers.
Warning! This is not the same as a power function.
(A function of the form f(x) = xn)
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Exponential Functions
Theorem. For an exponential function
f(x) = ax, a > 0, a ≠ 1, if x is any real number, then
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-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Graphing Exponential Functions
Example.
Problem: Graph f(x) = 3x
Answer:
![Page 42: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/42.jpg)
Graphing Exponential Functions
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Properties of the Exponential Function
Properties of f(x) = ax, a > 1
Domain: All real numbers
Range: Positive real numbers; (0, ∞)
Intercepts:No x-intercepts
y-intercept of y = 1
x-axis is horizontal asymptote as x → —∞
Increasing and one-to-one.
Smooth and continuous
Contains points (0,1), (1, a) and
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Properties of the Exponential Function
f(x) = ax, a > 1
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Properties of the Exponential Function
Properties of f(x) = ax, 0 < a < 1
Domain: All real numbers
Range: Positive real numbers; (0, ∞)
Intercepts:No x-intercepts
y-intercept of y = 1
x-axis is horizontal asymptote as x → ∞
Decreasing and one-to-one.
Smooth and continuous
Contains points (0,1), (1, a) and
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Properties of the Exponential Function
f(x) = ax, 0 < a < 1
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The Number e
Number e: the number that the expression
approaches as n → ∞.
Use ex or exp(x) on your calculator.
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The Number e
Estimating value of e
n = 1: 2
n = 2: 2.25
n = 5: 2.488 32
n = 10: 2.593 742 460 1
n = 100: 2.704 813 829 42
n = 1000: 2.716 923 932 24
n = 1,000,000,000: 2.718 281 827 10
n = 1,000,000,000,000: 2.718 281 828 46
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Exponential Equations
If au = av, then u = v
Another way of saying that the function f(x) = ax is one-to-one.
Examples.
(a) Problem: Solve 23x —1 = 32
Answer:
(b) Problem: Solve
Answer:
![Page 50: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/50.jpg)
Key Points
Exponents
Laws of Exponents
Exponential Functions
Graphing Exponential Functions
Properties of the Exponential Function
The Number e
Exponential Equations
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Logarithmic Functions
Section 4.4
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Logarithmic Functions
Logarithmic function to the base a
a > 0 and a ≠ 1
Denoted by y = logax
Read “logarithm to the base a of x” or “base a logarithm of x”Defined: y = logax if and only if x = ay
Inverse function of y = ax
Domain: All positive numbers (0,∞)
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Logarithmic Functions
Examples. Evaluate the following logarithms
(a) Problem: log7 49
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
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Logarithmic Functions
Examples. Change each exponential expression to an equivalent expression involving a logarithm
(a) Problem: 2π = s
Answer:
(b) Problem: ed = 13
Answer:
(c) Problem: a5 = 33
Answer:
![Page 55: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/55.jpg)
Logarithmic Functions
Examples. Change each logarithmic expression to an equivalent expression involving an exponent.
(a) Problem: loga 10 = 7
Answer:
(b) Problem: loge t = 4
Answer:
(c) Problem: log5 17 = z
Answer:
![Page 56: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/56.jpg)
Domain and Range of Logarithmic Functions
Logarithmic function is inverse of the exponential function.
Domain of the logarithmic functionSame as range of the exponential function
All positive real numbers, (0, ∞)
Range of the logarithmic functionSame as domain of the exponential function
All real numbers, (—∞, ∞)
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Domain and Range of Logarithmic Functions
Examples. Find the domain of each
function
(a) Problem: f(x) = log9(4 — x2)
Answer:
(b) Problem:
Answer:
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-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Graphing Logarithmic Functions
Example. Graph the function
Problem: f(x) = log3 x
Answer:
![Page 59: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/59.jpg)
Properties of the Logarithmic Function
Properties of f(x) = loga x, a > 1Domain: Positive real numbers; (0, ∞)
Range: All real numbers
Intercepts:x-intercept of x = 1
No y-intercepts
y-axis is horizontal asymptote
Increasing and one-to-one.
Smooth and continuous
Contains points (1,0), (a, 1) and
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Properties of the Logarithmic Function
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Properties of the Logarithmic Function
Properties of f(x) = loga x, 0 < a < 1Domain: Positive real numbers; (0, ∞)
Range: All real numbers
Intercepts:x-intercept of x = 1
No y-intercepts
y-axis is horizontal asymptote
Decreasing and one-to-one.
Smooth and continuous
Contains points (1,0), (a, 1) and
![Page 62: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/62.jpg)
Properties of the Logarithmic Function
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Special Logarithm Functions
Natural logarithm:
y = ln x if and only if x = ey
ln x = loge x
Common logarithm:
y = log x if and only if x = 10y
log x = log10 x
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-6 -4 -2 2 4 6
-6
-4
-2
2
4
6
Special Logarithm Functions
Example. Graph the function
Problem: f(x) = ln (3—x)
Answer:
![Page 65: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/65.jpg)
Logarithmic Equations
Examples. Solve the logarithmic equations. Give exact answers.
(a) Problem: log4 x = 3
Answer:
(b) Problem: log6(x—4) = 3
Answer:
(c) Problem: 2 + 4 ln x = 10
Answer:
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Logarithmic Equations
Examples. Solve the exponential equations using logarithms. Give exact answers.
(a) Problem: 31+2x= 243
Answer:
(b) Problem: ex+8 = 3
Answer:
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Key Points
Logarithmic Functions
Domain and Range of Logarithmic Functions
Graphing Logarithmic Functions
Properties of the Logarithmic Function
Special Logarithm Functions
Logarithmic Equations
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Properties of Logarithms
Section 4.5
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Properties of Logarithms
Theorem. [Properties of Logarithms]
For a > 0, a ≠ 1, and r some real
number:
loga 1 = 0
loga a = 1
loga ar = r
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Properties of Logarithms
Theorem. [Properties of Logarithms]
For M, N, a > 0, a ≠ 1, and r some real number:
loga (MN) = loga M + loga N
loga Mr = r loga M
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Properties of Logarithms
Examples. Evaluate the following expressions.
(a) Problem:
Answer:
(b) Problem: log140 10 + log140 14
Answer:
(c) Problem: 2 ln e2.42
Answer:
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Properties of Logarithms
Examples. Evaluate the following expressions if logb A = 5 and logbB = —4.
(a) Problem: logb AB
Answer:
(b) Problem:
Answer:
(c) Problem:
Answer:
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Properties of Logarithms
Example. Write the following expression as a sum of logarithms. Express all powers as factors.
Problem:
Answer:
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Properties of Logarithms
Example. Write the following expression as a single logarithm.
Problem: loga q — loga r + 6 loga p
Answer:
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Properties of Logarithms
Theorem. [Properties of Logarithms]
For M, N, a > 0, a ≠ 1,If M = N, then loga M = loga N
If loga M = loga N, then M = N
Comes from fact that exponential and logarithmic functions are inverses.
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Logarithms with Bases Other than e and 10
Example.
Problem: Approximate log3 19 rounded to four decimal places
Answer:
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Logarithms with Bases Other than e and 10
Theorem. [Change-of-Base Formula]
If a ≠ 1, b ≠ 1 and M are all positive real numbers, then
In particular,
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Logarithms with Bases Other than e and 10
Examples. Approximate the following logarithms to four decimal places
(a) Problem: log6.32 65.16
Answer:
(b) Problem:
Answer:
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Key Points
Properties of Logarithms
Properties of Logarithms
Logarithms with Bases Other than eand 10
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Logarithmic and Exponential Equations
Section 4.6
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Solving Logarithmic Equations
Example. Problem: Solve log3 4 = 2 log3 x
algebraically.
Answer:
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Solving Logarithmic Equations
Example.
Problem: Solve log3 4 = 2 log3 xgraphically.
Answer:
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Solving Logarithmic Equations
Example. Problem: Solve log2(x+2) + log2(1—x) = 1
algebraically.
Answer:
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Solving Logarithmic Equations
Example.
Problem: Solve log2(x+2) + log2(1—x) = 1 graphically.
Answer:
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Solving Exponential Equations
Example. Problem: Solve 9x — 3x — 6 = 0
algebraically.
Answer:
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Solving Exponential Equations
Example.
Problem: Solve 9x — 3x — 6 = 0 graphically.
Answer:
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Solving Exponential Equations
Example. Problem: Solve 3x = 7 algebraically. Give
an exact answer, then approximate your answer to four decimal places.
Answer:
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Solving Exponential Equations
Example.
Problem: Solve 3x = 7 graphically. Approximate your answer to four decimal places.
Answer:
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Solving Exponential Equations
Example. Problem: Solve 5 · 2x = 3 algebraically.
Give an exact answer, then approximate your answer to four decimal places.
Answer:
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Solving Exponential Equations
Example.
Problem: Solve 5 · 2x = 3 graphically. Approximate your answer to four decimal places.
Answer:
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Solving Exponential Equations
Example. Problem: Solve 2x—1 = 52x+3 algebraically.
Give an exact answer, then approximate your answer to four decimal places.
Answer:
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Solving Exponential Equations
Example.
Problem: Solve e2x — x2 = 3 graphically. Approximate your answer to four decimal places.
Answer:
![Page 93: Exponential and Logarithmic Functions€¦ · Exponential Functions zExponential function: function of the form f(x) = ax zwhere a is a positive real number (a>0) za ≠1. zDomain](https://reader035.vdocuments.site/reader035/viewer/2022062602/5edc80f0ad6a402d666730ac/html5/thumbnails/93.jpg)
Key Points
Solving Logarithmic Equations
Solving Exponential Equations
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Compound Interest
Section 4.7
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Simple Interest
Simple Interest Formula
Principal of P dollars borrowed for tyears at per annum interest rate r
Interest is I = Prt
r must be expressed as decimal
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Compound Interest
Payment period
Annually: Once per year
Semiannually: Twice per year
Quarterly: Four times per year
Monthly: 12 times per year
Daily: 365 times per year
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Compound Interest
Theorem. [Compound Interest Formula]The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is
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Compound Interest
Example. Find the amount that results from the investment of $1000 at 8% after a period of 8 years.
(a) Problem: Compounded annually
Answer:
(b) Problem: Compounded quarterly
Answer:
(c) Problem: Compounded daily
Answer:
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Compound Interest
Theorem. [Continuous Compounding]The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is
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Compound Interest
Example. Find the amount that results from the investment of $1000 at 8% after a period of 8 years.
Problem: Compounded continuously
Answer:
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Effective Rates of Interest
Effective Rate of Interest: Equivalent annual simple interest rate that yields same amount as compounding after 1 year.
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Effective Rates of Interest
Example. Find the effective rate of interest on an investment at 8%
(a) Problem: Compounded monthly
Answer:
(a) Problem: Compounded daily
Answer:
(a) Problem: Compounded continuously
Answer:
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Present Value
Present value: amount needed to invest now to receive A dollars at a specified future time.
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Present Value
Theorem. [Present Value Formulas]The present value P of A dollars to be received after t years, assuming a per annum interest rate rcompounded n times per year, is
if the interest is compounded continuously, then
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Present Value
Example.
Problem: Find the present value of $5600 after 4 years at 10% compounded semiannually. Round to the nearest cent.
Answer:
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Time to Double an Investment
Example.
Problem: What annual rate of interest is required to double an investment in 8 years?
Answer:
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Key Points
Simple Interest
Compound Interest
Effective Rates of Interest
Present Value
Time to Double an Investment
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Exponential Growth and Decay; Newton’s Law; Logistic Growth and Decay
Section 4.8
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Uninhibited Growth and Decay
Uninhibited Growth:
No restriction to growth
Examples
Cell division (early in process)
Compound Interest
Uninhibited Decay
Examples
Radioactive decayCompute half-life
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Uninhibited Growth and Decay
Uninhibited Growth:
N(t) = N0 ekt, k > 0
N0: initial population
k: positive constant
t: time
Uninhibited Decay
A(t) = A0 ekt, k < 0
N0: initial amount
k: negative constant
t: time
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Uninhibited Growth and Decay
Example.
Problem: The size P of a small herbivore population at time t (in years) obeys the function P(t) = 600e0.24t if they have enough food and the predator population stays constant. After how many years will the population reach 1800?
Answer:
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Uninhibited Growth and Decay
Example.
Problem: The half-life of carbon 14 is 5600 years. A fossilized leaf contains 12% of its normal amount of carbon 14. How old is the fossil (to the nearest year)?
Answer:
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Newton’s Law of Cooling
Temperature of a heated object decreases exponentially toward temperature of surrounding medium
Newton’s Law of CoolingThe temperature u of a heated object at a given time t can be modeled by
u(t) = T + (u0 — T)ekt, k < 0
where T is the constant temperature of the surrounding medium, u0 is the initial temperature of the heated object, and k is a negative constant.
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Newton’s Law of Cooling
Example.
Problem: The temperature of a dead body that has been cooling in a room set at 70◦F is measured as 88◦F. One hour later, the body temperature is 87.5◦F. How long (to the nearest hour) before the first measurement was the time of death, assuming that body temperature at the time of death was 98.6◦F?
Answer:
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Logistic Model
Uninhibited growth is limited in actuality
Growth starts off like exponential, then levels off
This is logistic growth
Population approaches carrying capacity
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Logistic Model
Logistic ModelIn a logistic growth model, the population P after time t obeys the equation
where a, b and c are constants with c > 0 (c is the carrying capacity). The model is a growth model if b > 0; the model is a decay model if b < 0.
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Logistic Model
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Logistic Model
Properties of Logistic Function
Domain is set of all real numbers
Range is interval (0, c)
Intercepts:
no x-intercept
y-intercept is P(0).
Increasing if b > 0, decreasing if b < 0
Inflection point when P(t) = 0.5c
Graph is smooth and continuous
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Logistic Model
Example. The logistic growth model
represents the population of a species introduced into a new territory after tyears.(a) Problem: What was the initial population
introduced?Answer:
(b) Problem: When will the population reach 80?Answer:
(c) Problem: What is the carrying capacity?Answer:
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Key Points
Uninhibited Growth and Decay
Newton’s Law of Cooling
Logistic Model
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Building Exponential, Logarithmic, and Logistic Models from Data
Section 4.9
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Fitting an Exponential Function to Data
Example. The population (in hundred thousands) for the Colonial US in ten-year increments for the years 1700-1780 is given in the following table. (Source: 1998 Information Please Almanac)
Decade, x Population, P
0 251
1 332
2 466
3 629
4 906
5 1171
6 1594
7 2148
8 2780
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Fitting an Exponential Function to Data
Example. (cont.)
(a) Problem: State whether the data can be more accurately modeled using an exponential or logarithmic function.
Answer:
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Fitting an Exponential Function to Data
Example. (cont.)
(b) Problem: Find a model for population (in hundred thousands) as a function of decades since 1700.
Answer:
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Fitting a Logarithmic Function to Data
Example. The death rate (in deaths per 100,000 population) for 20-24 year olds in the US between 1985-1993 are given in the following table. (Source: NCHS Data Warehouse)
Year Rate of Death, r
1985 134.9
1987 154.7
1989 162.9
1991 174.5
1992 182.2
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Fitting a Logarithmic Function to Data
Example. (cont.)
(a) Problem: Find a model for death rate in terms of x, where x denotes the number of years since 1980.
Answer:
(b) Problem: Predict the year in which the death rate first exceeded 200.
Answer:
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Fitting a Logistic Function to Data
Example. A mechanic is testing the cooling system of a boat engine. He measures the engine’s temperature over time.
Time t(min.)
Temperature T(◦F)
5 100
10 180
15 270
20 300
25 305
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Fitting a Logistic Function to Data
Example. (cont.)
(a) Problem: Find a model for the
temperature T in terms of t, time in
minutes.
Answer:
(b) Problem: What does the model imply
will happen to the temperature as time
passes?
Answer:
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Key Points
Fitting an Exponential Function to Data
Fitting a Logarithmic Function to Data
Fitting a Logistic Function to Data