5.8 Inverse Functions and Logarithms 5.8I nverse Functions and Logarithms This is Section 5.5 in the current text In this section, students will learn about the properties and characteristics of exponential functions. After completing this lesson the students will be able to: • Justify whether or not a function is one-to-one. • Decide whether or not two functions are inverses of each other. • Identify a logarithmic function. • Convert equations between exponential and logarithmic form. • Memorize the graph of the parent logarithmic function, base a. • Determine the domain of a logarithmic function, using interval notation. • Apply the laws and properties of logarithms to expand, condense, and simplify logarithmic expressions. • Solve equations involving exponential functions with different bases. • Solve equations involving logarithms. • Use exponential and logarithmic functions to model and solve real-world applications. Defining an Inverse Function Definition Suppose f and g are two function such that 1. (g ◦ f )(x) = x for all x in the domain of f and 2. ( f ◦ g)(x) = x for all x in the domain of g then f and g are inverses of each other and the functions f and g are said to be invertible. Properties of Inverse Functions Suppose f and g are inverse functions. 1. There is exactly one inverse function for f denoted f −1(x) = g(x). 2. f (a) = b if and only if g(b) = a 3. (a;b) is on the graph of f if and only if (b;a) is on the graph of g 4. The range of f is the domain of g and the domain of f is the range of g 5. The graph of g(x) = f −1(x) is the reflection of the graph of f (x) across the line y = x. −1 1 ! The notation f is an unfortunate choice since you want to think of this as f . This is definitely not the case x − 4 since, for instance, f (x) = 3x + 4 has as its inverse f −1(x) = , which is certainly different than 3 1 1 = . f (x) 3x + 4 Besides using compositions, one way to determine if a function is invertible is to determine if the function is one-to-one. Definition A function f is said to be one-to-one if f matches different inputs to different outputs. Equivalently f is one-to-one if and only if whenever f (c) = f (d), then c = d (or if c , d then f (c) , f (d)). 169 © TAMU Graphically, we detect one-to-one functions using the test below. Theorem 5.1 The Horizontal Line Test: A function f is one-to-one if and only if no horizontal line intersects the graph of f more than once. We say that the graph of a function passes the Horizontal Line Test if no horizontal line intersects the graph more than once; otherwise we say the graph of the function fails the Horizontal Line Test. Theorem 5.2 Equivalent conditions for Invertibility: Suppose f (X) is a function. The following statements are equivalent. • f (x) is invertible. • f (x) is one-to-one. • The graph of f (x) passes the Horizontal Line Test. p 3 x + 4 Example1 Determine if g(x) = is one-to-one using the Horizontal Line Test. 5 − 3x p 2 Example2 Determine whether or not f (x) = x + 7 for x ≥ 0 and g(x) = x − 7 are inverses of each other. © TAMU 170 5.8 Inverse Functions and Logarithms Defining Logarithmic Functions Definition In general, the inverse of the exponential function f (x) = bx is called the base b logarithm function, and is −1 denoted f (x) = logb(x). We have special notations for the common base, b = 10, and the natural base b = e. Definition • The common logarithm of a real number x is log10(x) and is usually written log(x) • The natural logarithm of a real number x is loge(x) and is usually written ln(x) The following defines a logarithmic function using exponent notation instead of inverse notation. Definition A logarithm base b of a positive number x satisfies the following definition. For x > 0, b > 0, b , 1, y y = logb(x) is equivalent to b = x where, • we read logb(x) as the “logarithm with base b of x” or the “log base b of x”. • the logarithm y is the exponent to b must be raised to get x. Properties of Logarithmic Functions Recall that the exponential function is defined as y = bx for any real number x and constant b > 0, b , 1, where • The domain of y is (−∞;1). • The range of y is (0;1). x We just learned that the logarithmic function y = logb(x) is the inverse of the exponential function y = b . So, as the inverse function: x • The domain of y = logb(x) is the range of y = b : (0;1). x • The range of y = logb(x) is is the domain of y = b :(−∞;1). 171 © TAMU Properties of Logarithmic Functions Suppose f (x) = logb(x). Inverse Properties • The domain of f (x) is 0;1. • The range of f (x) is −∞;1. • The x-intercept is (1,0). • There is no y-intercept. • f (x) is one-to-one. • If b > 1 • If 0 < b < 1 - End behavior as x ! 0 from - End behavior as x ! 0 from the the right, f (x) ! −∞ (vertical right, f (x) ! 1 (vertical asymp- asymptote x = 0) tote x = 0) - End behavior as x ! 1 , f (x) ! - End behavior as x ! 1 , f (x) ! 1 −∞ - The graph of f resembles: - The graph of f resembles: 10 10 5 5 x x −10 −5 5 10 −10 −5 5 10 −5 −5 − − 10 y 10 y Exponent Properties a • b = c if and only if logb(c) = a. That is, logb(c) is the exponent you put on b to get c. x logb(x) • logb(b ) = x for all x and b = x for all x > 0 We will add our final two parent functions to our list of parent functions. Name Function Graph Domain 10 f (x) = log (x) Logarithmic Growth b x (0;1) b > 1 −10 −10 10 y f (x) = log (x) Logarithmic Decay b x (0;1) 0 < b < 1 −10 − 10 y © TAMU 172 5.8 Inverse Functions and Logarithms Example3 Given h(x) = log 2 (x), state each of the following: 3 a. Domain b. Range c. End behavior d. x-intercept(s) e. y-intercept(s) Computing the Domain of a Logarithmic Function Up until this point, restrictions on the domains of functions came from avoiding division by zero and keeping negative numbers from beneath even radicals. With the introduction of logs, we now have another restriction. Since the domain of f (x) = logb(x) is (0;1), the input argument of the logarithm must be strictly positive. Generally speaking, in order for f (x) = logb(g(x)) to be defined g(x) must be defined and g(x) must be greater than zero. Example4 State the domain of the following functions, using interval notation. a. f (x) = ln(x − 1) + 4 ! 3 b. h(x) = log x + 8 173 © TAMU Using Algebraic Properties of Logarithms We introduced logarithmic functions as inverse of exponential functions and discussed a few of their functional properties from that perspective. In this section, we explore the algebraic properties of logarithms. Historically, these have played a huge role in the scientific development of our society since, among other things, they were used to develop analog computing devices called slide rules which enabled scientists and engineers to perform accurate calculations leading to such things a space travel and the moon landing. Converting Equations between Exponential and Logarithmic Forms Example5 Write the following logarithmic equations in exponential form. p 1 a. ln e = 2 b. log 1 (4) = −2 2 Example6 Write the following exponential equations in logarithmic form. 1 a. 2−5 = 32 !3 1 1 b. = 5 125 © TAMU 174 5.8 Inverse Functions and Logarithms Theorem 5.3 Change of Base Formula Let a;b > 0, a , 1, b , 1, x > 0 logb(x) log(x) ln(x) • loga(x) = for all real numbers x > 0. In particular loga(x) = = logb(a) log(a) ln(a) Condensing and Expanding Logarithms Returning to the concept that the logarithmic and exponential functions “undo” each other, this means that logarithms have similar properties to exponentials. Some important properties of logarithms are given here. Let g(x) = logb(x) be a logarithmic function (b > 0;b , 1) and let M > 0 and N > 0 be real numbers. • g(1) = logb(1) = 0 • g(b) = logb(b) = 1 • Product Rule logb(MN) = logb(M) + logb N M • Quotient Rule log = log (M) − log N b N b b N • Power Rule logb(M ) = N logb M ­ ln(e) = 1 and log(10) = 1 M logb(M) ! logb , N logb(N) N n ! logb(M) , N logb(M) logb(ax ) , nlogb(ax) Example7 Use the properties of logarithms to write the following as a single logarithm for x;y;z > 0. 2ln(x) + ln(y + 6) − 4z 175 © TAMU Example8 Use the properties of logarithms to write the following as a single logarithm for −4 < x < 7. 1 log (7 − x) + log (x + 4) − 6log (x) 7 3 7 7 Example9 Expand the following using the properties of logarithms and simplify. Assume when necessary that all quantities represent positive real numbers. log x2z Example 10 Expand the following using the properties of logarithms and simplify.