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Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Section 9.1: Exponential Functions
Definition of exponential function
The exponential function f with base a is denoted by
f (x) = ax
where a > 0, a 6= 1, and x is any real number.
Rules of exponential functions
Let a be a positive real number, and let x and y be real numbers variables, or algebraic expressions.
(1) ax · ay = ax+y
x
(2) aay = ax−y
(3) (ax )y = axy
(4) a−x = a1x = ( a1 )x
Ex.1
Evaluate f (x) = 2x when x = 3 and when x = −2.
Definition of asymptote
An asymptote of a graph is a line to which the graph becomes arbitrarily close as |x| or |y| increases without
bound.
1
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Ex.2
In the same coordinate plane, sketch the graph of each function. Determine the domain and the range.
(1) f (x) = 2x
(2) g(x) = 4x
Ex.3
In the same coordinate plane, sketch the graph of each function. Determine the domain and the range.
(1) f (x) = 2−x
(2) g(x) = 4−x
2
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Ex.4
A particular radioactive element has a half-life of 25 years. For an initial mass of 10 grams, the mass y (in
grams) that remains after t years is given by
1 t/25
, t≥0
y = 10
2
How much of the initial mass remains after 120 years?
Section 9.2: Composite and Inverse Functions
Definition of composition of two functions
The composition of the functions f and g is given by
(f ◦ g)(x) = f (g(x))
The domain of the composite function (f ◦ g) is the set of all x in the domain of g such that g(x) is in the
domain of f .
3
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Ex.1
Given f (x) = 2x + 4 and g(x) = 3x − 1, find the composition of f with g. Then evaluate the composite
function when x = 1 and when x = −3.
Ex.2
Given f (x) = 2x − 3 and g(x) = x2 + 1, find each composition.
(1) (f ◦ g)(x)
(2) (g ◦ f )(x)
4
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Ex.3
√
Find the domain of the composition of f with g when f (x) = x2 and g(x) = x.
Definition of inverse function
Let f and g be two functions such that f (g(x)) = x for every x in the domain of g and g(f (x)) = x for every
x in the domain of f . The function g is called the inverse function of the function f , and is denoted by f −1 .
The domain of f must be equal to the range of f −1 . The graph of f −1 is the symmetric graph of f (x) with
respect to the line y = x
5
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Definition of one-to-one function
A function f is one-to-one if each value of the dependent variable corresponds to exactly one value of the
independent variable.
Horizontal line test for inverse functions
A function f has an inverse function f −1 if and only if f is one-to-one. Graphically, a function f has an
inverse function if and only if no horizontal line intersects the graph of f at more than one point.
Ex.4
The function f (x) = x2 is not one-to-one.
Ex.5
Show that f (x) = 2x − 4 and g(x) =
x+4
2
are inverse functions of each other.
6
Chapter 9: Exponential and Log. Functions
Lecture notes
Ex.6
√
Show that f (x) = x3 + 1 and g(x) = 3 x − 1 are inverse functions of each other.
7
Math 1010
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Finding an inverse function algebraically
(1) In the equation for f (x), replace f (x) with y.
(2) Interchange x and y.
(3) If the new equation does not represent y as a function of x, the function f does not have an inverse
function. If the new equation does represent y as a function of x, solve the new equation for y.
(4) Replace y with f −1 (x).
(5) Verify that f and f −1 are inverse functions of each other by showing that f (f −1 (x)) = x =
f −1 (f (x)).
Ex.7
Determine whether each function has an inverse function. If it does, find the inverse function.
(1) f (x) = 2x + 3
(2) f (x) = x3 + 3
(3) f (x) = x2
8
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Section 9.3: Logarithmic Functions
Definition of logarithmic function
Let a and x be positive real numbers such that a 6= 1. The logarithm of x with base a is denoted by loga x and
is defined as the power to which a must be raised to obtain x. The function f (x) = loga x is the logarithmic
function with base a .
Ex.1
Evaluate each logarithm.
(1) log2 8
(2) log2 16
(3) log3 9
(4) log4 2
9
Chapter 9: Exponential and Log. Functions
Lecture notes
Ex.2
Evaluate each logarithm.
(1) log5 1
1
(2) log10 10
(3) log3 (−1)
(4) log4 0
Definition of natural logarithmic function
The function f (x) = loge x = ln x, where x > 0 is the natural logarithmic function .
Properties of logarithms
Let a and x be positive real numbers such that a 6= 1. The following properties are true.
(1) loga 1 = 0 and ln 1 = 0.
(2) loga a = 1 and ln e = 1.
(3) loga ax = x and ln ex = x.
10
Math 1010
Chapter 9: Exponential and Log. Functions
Lecture notes
Ex.3
Evaluate each logarithm.
(1) log10 100
(2) log10 0.01
(3) ln e2
(4) ln 1e
Change-of-base formula
Let a, b and x be positive real numbers such that a 6= 1 and b 6= 1. Then
ln x
logb x
or loga x =
loga x =
logb a
ln a
Ex.4
(1) Use logarithms in base 10 to evaluate log3 5.
(2) Use natural logarithms to evaluate log6 2.
11
Math 1010
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Section 9.4: Properties of Logarithms
Properties of logarithms
Let a be a positive real number such that a 6= 1, and let n be a real number. If u and v are real numbers,
variables, or algebraic expressions such that u > 0 and v > 0, the following properties are true.
(1) loga (uv) = loga u + loga v and ln(uv) = ln u + ln v.
(2) loga ( uv ) = loga u − loga v and ln( uv ) = ln u − ln v.
(3) loga un = n loga u and ln un = n ln u.
Ex.1
Use ln 2 ∼ 0.693, ln 3 ∼ 1.099, and ln 5 ∼ 1.609 to approximate each expression.
(1) ln 23
(2) ln 10
(3) ln 30
12
Chapter 9: Exponential and Log. Functions
Lecture notes
Ex.2
Use the properties of logarithms to verify that − ln 2 = ln 12 .
Ex.3
Use the properties of logarithms to expand each expression.
(1) log10 7x3
3
(2) log6 8xy
√
(3) ln
3x−5
7
Ex.4
Use the properties of logarithms to condense each expression.
(1) ln x − ln 3
(2) 12 log3 x + log3 5
(3) 3(ln 4 + ln x)
13
Math 1010
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Section 9.5: Solving Exponential and Logarithmic Equations
Properties of exponential and logarithmic equations
Let a be a positive real number such that a 6= 1, and let x and y be real numbers. Then the following
properties are true.
(1) ax = ay if and only if x = y.
(2) loga x = loga y if and only if x = y (x > 0, y > 0).
(3) loga (ax ) = x and ln(ex ) = x.
(4) a(loga x) = x and e(ln x) = x.
Ex.1
Solve each equation.
(1) 4x+2 = 64
(2) ln(2x − 3) = ln 11
Solving exponential equations
To solve an exponential equation, first isolate the exponential expression. Then take the logarithm of each
side of the equation and solve for the variable.
14
Chapter 9: Exponential and Log. Functions
Lecture notes
Ex.2
Solve each exponential equation.
(1) 2x = 7
(2) 4x−3 = 9
(3) 2ex = 10
(4) 5 + ex+1 = 20
15
Math 1010
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Solving logarithmic equations
To solve a logarithmic equation, first isolate the logarithmic expression. Then exponentiate each side of the
equation and solve for the variable.
Ex.3
Solve each logarithmic equation.
(1) 2 log4 x = 5
(2) 14 log2 x = 21
(3) 3 log10 x = 6
(4) log3 2x − log3 (x − 3) = 1
16
Chapter 9: Exponential and Log. Functions
Lecture notes
Math 1010
Ex.4
A deposit of $5000 is placed in a savings account for 2 years. The interest on the account is compounded
continuously. At the end of 2 years, the balance in the account is $5416.44. What is the annual interest
rate for this account?
17
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