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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