Exponential and Logarithmic Functions Chapter 9 Exponential and Logarithmic Functions

The Algebra of Functions; Composite Functions § 9.1 The Algebra of Functions; Composite Functions

Operations on Functions It is possible to add, subtract, multiply, and divide functions. The results of these operations will also be functions (assuming we don’t divide by zero).

Operations on Functions Algebra of Functions Let f and g be functions. New functions from f and g are defined as follows: Sum (f + g)(x) = f(x) + g(x) Difference (f – g)(x) = f(x) – g(x) Product (f · g)(x) = f(x) · g(x) Quotient

Operations on Functions Example: If f(x) = 4x + 3 and g(x) = x2, then find each of the following (f + g)(x) 4x + 3 + x2 = x2 + 4x + 3 (f – g)(x) 4x + 3 – x2 = -x2 + 4x + 3 (f · g)(x) (4x + 3)x2 = 4x3 + 3x2

Operations on Functions Example: If f(x) = 4x + 3 and g(x) = x2, then find x  0

Function Composition We can also combine functions through a function composition. A function composition uses the output from the first function as the input to the second function. Composition of a Function The composition of function f and g is This means the value of x is first substituted into the function g. Then the value that results from the function g is input into the function f.

Function Composition Notice, that with function composition, we actually activate the functions from right to left in the notation. The function named on the right side of the composition notation is the one we substitute the value for the variable into first.

Function Composition Example: If f(x) = 4x + 3 and g(x) = x2, then find f(g(x)) = 4(x2) + 3 = 4x2 + 3 g(f(x)) = (4x + 3)2 = 16x2 + 24x + 9 Notice the results are different with a different order.

Function Composition Example: If H(x) = x3 + 3, name two functions whose composition will result in H(x). Note: There may be more than one way to select the two functions. Answers are not necessarily unique. Let f(x) = x + 3, and g(x) = x3

§ 9.2 Inverse Functions

One-to-One Functions We have studied functions which are defined to require that each element of the domain (input values) produce a unique element of the range (output values). 1-1 functions also require that each element of the range be produced from only one element of the domain. each x  only one y is a function each y  only one x is a 1-1 function

One-to-One Functions Example: Determine whether each function described is one to-one. r = {(1, 2), (3, 4), (5, 6), (6, 7)} It is 1-to-1, since the 2nd coordinate is unique. g = {(0, 3), (3, 7), (6, 7), (-2, -2)} It is not 1-to-1, since the second coordinate, 7, is produced from two different domain elements, both 3 and 6.

One-to-One Functions We found previously that the graph of a function must satisfy the Vertical Line Test. Every vertical line must intersect the graph of the function at most one time. Similarly, a 1-to-1 function must satisfy the Horizontal Line Test. Horizontal Line Test Every horizontal line must intersect the graph of the 1-to-1 function at most one time.

One-to-One Functions Example: Is this the graph of a 1-to-1 function? y Is this the graph of a 1-to-1 function? No, because we can find a horizontal line that intersects the graph in more than one point.

One-to-One Functions Example: Is this the graph of a 1-to-1 function? y Is this the graph of a 1-to-1 function? Yes, because every horizontal line intersects the graph in only one point.

One-to-One Functions Note that while we previously discovered that a horizontal line represents a function, it is NOT a 1-to-1 function. It will not pass the Horizontal Line Test.

Inverse Functions For each 1-to-1 function, we can find an inverse function by switching the coordinates of the ordered pairs of the original function. For a particular function f, the notation for the inverse function is f –1, and is read “f inverse”. Note that since we switch the coordinates in the ordered pairs to find the inverse function, the domain of f is the range of f -1, and the range of f is the domain of f -1. So, for the 1-to-1 function f consisting of the ordered pairs (x, y), the inverse function f –1 consists of the ordered pairs (y, x).

Inverse Functions Example: Find the inverse of the 1-to-1 function g = {(3, 4), (6, 2), (2, -3), (4, -5)}. g-1 = {(4, 3), (2, 6), (-3, 2), (-5, 4)}

Inverse Functions If a 1-to-1 function is defined as a set of ordered pairs, it is relatively easy to find the inverse function (just switch the 1st and 2nd coordinates) Finding the Inverse of a One-to-One Function 1.) Replace f(x) with y. 2.) Interchange x and y. 3.) Solve the equation for y. 4.) Replace y with the notation f –1(x).

Inverse Functions Example: Find an equation of the inverse of f(x) = 6x – 1. Then graph both f and f –1 on the same set of axes. y = 6x – 1 x = 6y – 1 6y = x + 1  4) Continued.

Inverse Functions Example continued: f(x) = 6x – 1 y Notice that f and f –1 are mirror images of each other around the line y = x. A function and its inverse are symmetric about the line y = x.

Inverse Functions If f is a 1-to-1 function, then the inverse of f is the function f –1 such that and

Inverse Functions Example: If f(x) = x3 – 5, show that f –1(x) = Since both of the compositions give us x, is the inverse function of f(x) = x3 – 5.

Exponential Functions § 9.3 Exponential Functions

Exponential Expressions We have previously worked with exponential expressions, where the exponent was a rational number The expression bx can actually be defined for all real numbers, x, including irrational numbers. However, the proof of this would have to wait until a higher level math course.

Exponential Functions A function of the form f(x) = bx is called an exponential function if b > 0, b is not 1, and x is a real number.

Exponential Functions We can graph exponential functions of the form f(x) = 3x, g(x) = 5x or h(x) = (½)x by substituting in values for x, and finding the corresponding function values to get ordered pairs. We would find all graphs satisfy the following properties: 1-to-1 function y-intercept (0, 1) no x-intercept domain is (-, ) range is (0, )

Graphs of Exponential Functions y We would find a pattern in the graphs of all the exponential functions of the type bx, where b > 1.

Graphs of Exponential Functions y We would find a pattern in the graphs of all the exponential functions of the type bx, where 0 < b < 1.

Graphs of Exponential Functions y (h, 1) We would find a pattern in the graphs of all the exponential functions of the type bx-h, where b > 1. The graph has the same shape as the graph for bx, except it is shifted to the right h units.

Graphs of Exponential Functions y (-h, 1) We would find a pattern in the graphs of all the exponential functions of the type bx+h, where b > 1. The graph has the same shape as the graph for bx, except it is shifted to the left h units.

Uniqueness of bx Uniqueness of bx Let b > 0 and b  1. Then bx = by is equivalent to x = y. Example: Solve 6x = 36 6x = 62 x = 2

Solving Exponential Functions Example: Solve 92x+1 = 81 92x+1 = 92 2x + 1 = 2 2x = 1 x = ½

Solving Exponential Functions Example: Solve 3-3 = 32x –3 = 2x

Solving Exponential Functions Example: Solve 43x-6 = 322x (22)3x-6 = (25)2x (22)3x-6 = 210x 26x-12 = 210x 6x – 12 = 10x –12 = 4x x = –3

Applications of Exponential Functions Many applications use exponential functions of various types. Compound interest formulas are exponential functions used to determine the amount of money accumulated or borrowed. Exponential functions with negative exponents can be used to describe situations of decay, while those with positive exponents can be used to describe situations of growth.

Applications of Exponential Functions Example: Find the total amount invested in a savings account if $5000 was invested and earned 6% compounded monthly for 18 years. Round your answer to two decimal places. The formula that is used for calculating compound interest is where P is the initial principal invested, r is the interest rate, n is the number of times interest is compounded each year, t is the time of the investment (in years) and A is the amount of money in the account. Continued.

Applications of Exponential Functions Example continued: $

Applications of Exponential Functions Example: An accidental spill of 100 grams of radioactive material in a local stream has led to the presence of radioactive debris decaying at a rate of 5% each day. Find how much debris still remains after 30 days. The formula that would be used for this problem is where A is the amount of radioactive material to start, r is the rate of decay, t is the number of days and y is the amount of radioactive material after the time period. Continued.

Applications of Exponential Functions Example continued: (exact answer)

Logarithmic Functions § 9.4 Logarithmic Functions

Graph of a Exponential Function If we graph an exponential function where the base > 1, we get an increasing function, as shown below. x y

Graph of a Logarithmic Function We can graph the inverse of the function, as shown below. x y This inverse function is referred to as a logarithmic function.

Logarithmic Functions Logarithmic Definition If b > 0 and b ≠ 1, then y = logb x means x = by for every x > 0 and every real number y.

Writing Exponential Functions Example: Write each of the following as an exponential equation. log4 16 = 2 4² = 16 log8 ⅛ = –1 8–1 = ⅛ c) log3 = ½

Writing Logarithmic Functions Example: Write each of the following as a logarithmic equation. a) 54 = 625 log5 625 = 4 b) 2–3 = ⅛ log2 ⅛ = –3 c) 41/3 = log4 = ⅓

Values of Logarithmic Expressions Example: Find the value of each of the following logarithmic expressions. a) log2 32 Since 25 = 32, then log2 32 = 5 b) log5 Since 5–2 = , then log5 = –2 c) log4 2 Since 4½ = 2, then log4 2= ½

Solving Logarithmic Equations Example: Solve log3 1 = x for x. First we rewrite the equation as an exponential equation. 3x = 1 Since 30 = 1, then x = 0.

Solving Logarithmic Equations Example: Solve logx 81 = 4 for x. First we rewrite the equation as an exponential equation. x4 = 81 Since 34 = 81, then x = 3.

Solving Logarithmic Equations Example: Solve log6 x = 2 for x. First we rewrite the equation as an exponential equation. 62 = x Since 62 = 36, then x = 36.

Properties of Logarithms 1) logb 1 = 0 2) logb bx = x 3) b = x logb x

Properties of Logarithms Example: Simplify each of the following expressions 1) log4 46 From Property 2, log4 46 = 6. 2) 7 log7 –3 From Property 3, 7 = –3. log7 –3

Logarithmic Functions If x is a positive real number, b is a constant positive real number, and b is not 1, then a logarithmic function is a function that can be defined by f(x) = logb x The domain of f is the set of positive real numbers, and the range of f is the set of real numbers. A logarithmic function is an inverse function of an exponential function.

Graphs of Logarithmic Functions To graph a logarithmic function, we first write the equation in exponential notation. Then we find ordered pairs that satisfy the equation and plot their corresponding points.

Graphs of Logarithmic Functions Example: Graph y = log2 x. x y Write the exponential form 2y = x. Then choose y-values and find corresponding x-values. x y 4 2 2 1 1 ½ -1 ¼ -2

Graphs of Logarithmic Functions Example: Graph y = log½ x. x y Write the exponential form (½)y = x. Then choose y-values and find corresponding x-values. x y ¼ 2 ½ 1 1 2 -1 4 -2

Logarithmic Functions In general, from these two previous examples, we would find that for the logarithmic function f(x) = logb x, b > 0, b ≠ 1, The function is a 1-to-1 function. has a domain of (0, ∞). has a range (–∞, ∞). has an x-intercept of (1, 0). has no y-intercept.

Properties of Logarithms § 9.5 Properties of Logarithms

Properties of Logarithms This section examines several properties of logarithms that allow you to simplify expressions. Recall that a logarithm is an exponent, so logarithmic properties are restatements of exponential properties.

Product Property Product Property of Logarithm If x, y, and b are positive real numbers and b ≠ 1, then logb xy = logb x + logb y Note that although the property allows you to split the logarithm of a product into the sum of two logarithms, we usually use the property to consolidate a sum of logarithms into a single logarithm.

Product Property Example: Write each expression as a single logarithm. log7 x + log7 4 log7 4x logz 5 + logz 8 logz 40 log2 x + log2 10 + log2 (x + 3) log2 10x(x + 3)

Quotient Property Product Property of Logarithm If x, y, and b are positive real numbers and b ≠ 1, then logb = logb x – logb y Note that although the property allows you to split the logarithm of a quotient into the difference of two logarithms, we usually use the property to consolidate a difference of logarithms into a single logarithm.

Quotient Property Example: Write each expression as a single logarithm. log7 x – log7 4 log7 2) logz 40 – logz 8 logz 5 3) log4 x – log4 10 – log4 (x + 3)

Power Property Product Property of Logarithm If x, y, and b are positive real numbers and b ≠ 1, then logb xr = r logb x

Power Property Example: Use the power property to rewrite each expression. log7 x –3 –3 log7 x 2) log5

Using Logarithm Properties We often need to combine more than one of the logarithmic properties to simplify an expression. Example: Write the expression as a single logarithm. 2 log5 x + log5 4 – 3 log5 (x + 5)

Using Logarithm Properties Example: Write the expression as a sum or difference of multiples of logarithms.

Using Logarithm Properties Example: If logb 2 = 0.43 and logb 3 = 0.68, find each of the following. logb 16 logb 24 = 4 logb 2 = 4(0.43) = 1.72

Common Logarithms, Natural Logarithms, and Change of Base § 9.6 Common Logarithms, Natural Logarithms, and Change of Base

Common and Natural Logs There are two logarithmic bases that occur so frequently in applications that they are given special names. Common logarithms are logarithms to base 10. Natural logarithms are logarithms to base e (an irrational number which is approximately equal to 2.7183).

Common Logarithms log x means log10 x You can use a calculator to approximate common logarithms using the button below. LOG To find exact values of common logarithms, use the definition of logarithms to rewrite the expressions in exponential form to evaluate.

Common Logarithms Example: Find the exact value of each of the following logarithms. log 10,000 log 104 = 4 2) log 0.001 log 10 –3 = –3 log 10½ = ½

Common Logarithms Example: Solve the following equation for the variable. Give both an exact answer and an answer approximated to four decimal places. (exact answer) (approximate answer)

Applications with Logarithms One of the most popular uses of common logarithms involves the Richter scale for measuring the intensity of earthquakes. For R (magnitude of the earthquake), a (amplitude in micrometers of the vertical motion of the ground at the recording station), T (number of seconds between successive seismic waves), and B (adjustment factor that takes into account the weakening of the seismic wave as the distance increases from the epicenter of the earthquake), the formula is

Applications with Logarithms Example: Find the intensity R of an earthquake when the amplitude a is 300 micrometers, time T between waves is 2.5 seconds, and B is 2.6. Round the answer to one decimal place.

Natural Logarithms ln x means loge x You can use a calculator to approximate natural logarithms using the button below. LN To find exact values of common logarithms, use the definition of logarithms to rewrite the expressions in exponential form to evaluate.

Natural Logarithms Example: Find the exact value of each of the following logarithms. ln e4 4 ⅓

Solving Logarithmic Equations Example: Solve the following equation for the variable. Give both an exact answer and an answer approximated to four decimal places. (exact answer) (approximate answer)

Change of Base Change of Base If a, b, and c are positive real numbers and neither b nor c is 1, then

Change of Base Example: Approximate log5 to four decimal places.

Exponential and Logarithmic Equations and Applications § 9.7 Exponential and Logarithmic Equations and Applications

Logarithmic Property of Equality Let a, b, and c be real numbers such that logb a and logb c are real numbers and b is not 1. Then logb a = logb c is equivalent to a = c.

Logarithmic Property of Equality Example: Solve 6x+3 = 2. (exact answer) (approximate answer)

Solving Logarithmic Equations By using the definition of logarithms in terms of exponents and properties of logarithms, we can solve logarithmic equations.

Solving Logarithmic Equations Example: Solve 4x = 7. (exact answer) (approximate answer)

Solving Logarithmic Equations Example: Solve the following logarithmic equation.

Solving Logarithmic Equations Example: Solve the following logarithmic equation.

Solving Logarithmic Equations Example: Solve the following logarithmic equation.