PreCalculus Notes MAT 129 Chapter 6: Exponential and Logarithmic - PowerPoint PPT Presentation

PreCalculus Notes MAT 129 Chapter 6: Exponential and Logarithmic Functions David J. Gisch Department of Mathematics Des Moines Area Community College September 30, 2011

Chapter 6 1 Section 6.1: Composite Functions Section 6.2: One-to-One Functions; Inverse Functions Section 6.3: Exponential Functions Section 6.4: Logarithmic Functions Section 6.5: Properties of Logarithms Section 6.6: Logarithmic and Exponential Equations Section 6.7: Compound Interest Section 6.8: Exponential Growth and Decay; Newton’s Law; Etc Section 6.9: Building Exponential, Logarithmic, and Logistic Models from Data

Section 6.1: Composite Functions Summary We will what composite functions are and how to find their domain and range.

Definition Given two functions f and g , the composite function, denoted by f ◦ g (read as “ f composed of g ” or “ f of g ”), is defined by ( f ◦ g )( x ) = f ( g ( x )) The domain of f ◦ g is the set of all numbers x in the domain of g such that g ( x ) is in the domain of f .

Example Let f ( x ) = 2 x 2 − 3 and g ( x ) = 3 x . Find: (a) ( f ◦ g )(1) (b) ( f ◦ g )(3) (c) ( f ◦ g )( − 2)

Example Use the given functions to write one composite function. (a) Let f ( x ) = 2 x 2 − 3 and g ( x ) = 3 x ; ( f ◦ g )( x ). (b) Let f ( x ) = 2 x 2 − 3 and g ( x ) = 3 x ; ( g ◦ f )( x ). (c) Let f ( x ) = x − 4 x 2 +1 and g ( x ) = 3 x ; ( f ◦ g )( x ).

Finding the Domain of ( f ◦ g )( x ) You first exclude any elements that are not in the domain of g . 1 You find elements not in the domain of f and set g ( x ) equal to those 2 values and solve. The resulting solutions are values thet are also excluded from the domain.

Find the Domain Example (a) Let f ( x ) = 2 x 2 − 3 and g ( x ) = 3 x .Find the domain of ( f ◦ g )( x ) (b) Let f ( x ) = 2 x 2 − 3 and g ( x ) = 1 x . Find the domain of ( f ◦ g )( x ) (c) Let f ( x ) = √ x − 6 and g ( x ) = 2 x − 1 . Find the domain of ( f ◦ g )( x )

Example Given: f ( x ) = √ x − 4, g ( x ) = x 2 + 1, and h ( x ) = x − 1 x +1 , find each of the following and state the domain. (a) ( f ◦ g )( x ) (b) ( h ◦ g )( x ) (c) ( g ◦ f )( x )

Section 6.2: One-to-One Functions; Inverse Functions Summary We will find out what it means for a function to be one-to-one; find inverse functions by analyzing graphs and equations; and graph inverse functions.

Definition A function is one-to-one if for any two inputs in the domain they map to two different outputs in the range. That is, if x 1 � = x 2 are in the domain then f ( x 1 ) � = f ( x 2 ).

Theorem When looking at the graph of a functions f , if every horizontal line intersects the graph in at most one point then f is one-to-one.

Theorem If f is one-to-one, then f is said to have an inverse f − 1 such that if f ( x ) = y then f − 1 ( y ) = x. Therefore, if you know that the graph of f (a 1-1 function) contains the point (2 , − 3), then you know f − 1 contains the point ( − 3 , 2).

Theorem The graph of f and f − 1 are symmetric with respect to the line y = x. Example Given the graph, graph the inverse.

Theorem Given f and its inverse f − 1 then for all values x we have ( f − 1 ◦ f )( x ) = f − 1 ( f ( x )) = x ( f ◦ f − 1 )( x ) = f ( f − 1 ( x )) = x and Example Check if f ( x ) = 3 x − 4 and g ( x ) = 1 3 x + 4 3 are inverses.

Calculating Inverses To calculate the inverse of a function you swap the variable y and x and solve for y . Example Find the inverse of f ( x ) = 2 x − 10.

Calculating Inverses Example Find the inverse of f ( x ) = 2 x 2 − 10.

Calculating Inverses Example Find the inverse of f ( x ) = 2 x 2 − 10 if x ≥ 0.

Calculating Inverses Example Find the inverse of f ( x ) = 2 x +1 x +1 .

Section 6.3: Exponential Functions Summary We will evaluate exponential functions; graph them; and solve exponential equations.

Theorem Recall your laws of exponents. a m a n = a m + n ( a n ) m = a m · n a m a n = a m − n 1 m = 1 a 0 = 1 a − m = 1 a m Definition An exponential functions is a function of the form f ( x ) = a x where a is a positive real number. The domain of f is the set of all real numbers.

Theorem For an exponential function of the form f ( x ) = a x where a > 0 , and a � = 1 , then f ( x + 1) = a f ( x ) Example Let’s try it for f ( x ) = 4 x .

Graphs of Exponential Functions http://www.analyzemath.com/expfunction/expfunction.html Click here.

f ( x ) = a x , with a > 1

f ( x ) = a x , with 0 < a < 1

Example Graph f ( x ) = 2 − x − 3 using transformations.

The Number e Definition The number e is defined as the number such that 1 + 1 � n � e = lim n n →∞ To see what we mean lets evaluate this for a few values of n . n 1 2 10 100 100,000 1,000,000 (1 + 1 n ) n 2 2.25 2.593742 2.704814 2.718268 2.718280

The Graph of e Figure: Graph of y = e x

Solving Exponential Equations Example Solve 2 2 x − 1 = 32 Example Solve e 2 x − 1 = e − x � 4 1 � e 3 x ·

Solving Exponential Equations Example

Section 6.4: Logarithmic Functions Summary We will change exponential functions to logarithmic functions and vice versa; evaluate logarithmic functions; determine the domaina nd range; and solve logarithmic functions.

Definition A logarithmic function to the base a , where a > 0 and a � = 1, is denoted by y = log a x (read as “log base a of x ”) and is defined by x = a y y = log a x if and only if The domain of a logarithmic function of this form is x > 0.

Example Change to the alternate form. (a) For example, 3 4 = 81, so 4 = log 3 81. (b) y = log 3 x (c) y = log 5 x (d) y = log 2 32

Example Change to the alternate form. (a) 1 . 2 3 = m (b) e b = 9 (c) − 3 = log e x

Domain of a Logarithmic Function Note: The reason this is true is because the logarithmic function y = log a x is defined as the inverse function of y = a x . For any inverse f − 1 ( x ) the domain is the range of f ( x ). In this case f ( x ) is an exponential function which has the range of (0 , ∞ ). Domain Range f ( x ) = a x ( −∞ , ∞ ) (0 , ∞ ) f − 1 ( x ) = log a x (0 , ∞ ) ( −∞ , ∞ )

Example Find the domain of each logarithmic function. (a) f ( x ) = log 3 ( x − 2) � x +3 � (b) g ( x ) = log 2 x − 1 2 x 2 (c) h ( x ) = log 1

Graphs of Logarithmic Functions Recall that if f ( x ) contains the point ( a , b ), then f − 1 ( x ) contains the point ( b , a ). Thus, as logarithmic functions are inverses of exponential functions we can reflect (about y = x ) the graph an exponential function to create the graph of a logarithmic function.

Graphs of Logarithmic Functions

The Natural Logarithm Function Rather than write log e we have a special term for this and we call it the natural logarithm. x = e y y = ln x if and only if

Example (a) Find the domain of the logarithmic function f ( x ) = − ln( x − 3). (b) Graph f ( x ). (c) Determine the range and vertical asymptote. (d) Find f − 1 ( x ). (e) Use f − 1 ( x ) to find the range of f . (f) graph f − 1 ( x ).

Note: If we have a logarithm of base 10 we call this the common logarithm. In this case we do not write the number of the base. x = 10 y y = log x if and only if

Example (a) Find the domain of the logarithmic function f ( x ) = 3 log( x + 5). (b) Graph f ( x ). (c) Determine the range and vertical asymptote. (d) Find f − 1 ( x ). (e) Use f − 1 ( x ) to find the range of f . (f) graph f − 1 ( x ).

Solving Logarithmic Equations Example Solve each of the following (a) log 3 (4 x − 7) = 2 (b) log 2 (2 x + 1) = 3 (c) e 2 x +5 = 8 (d) ln( x + 2) = 5

Example The formula D = 5 e − 0 . 4 h can be used to find the number of milligrams D of a certain drug that is in a patient’s bloodstream h hours after the drug was administered. When the number of milligrams reaches 2, the drug is to be administered again. What is the time between injections?

Section 6.5: Properties of Logarithms Summary We will work with the properties of logarithms to rewrite logarithms in different forms and to evaluate logarithms whose base is neither 10 nor e.

Properties of Logarithms log a 1 = 0 (1) log a a = 1 (2) a log a M = M (3) log a a r = (4) r log a ( MN ) = log a M + log a N (5) M log a = log a M − log a N (6) N log a M r = r log a M (7)

Example √ x − 1), with x > 1, as the sum of logarithms. Write log 2 ( x 2 3

Example x 4 Write log 6 ( x 2 +3) 2 , with x � = 0, as the difference of logarithms.

Example Write ln x 3 √ x − 2 ( x +1) 2 , with x > 2, as the sum and difference of logarithms.

Example Write 3 ln 2 + ln( x 2 ) + 2 as a single logarithm.

Example Write 1 2 log a 4 − 2 log a 5 as a single logarithm.

Change of Base Formula Theorem If a � = 1 , b � = 1 and M are positive real numbers, then log a M = log b M log b a Example Evaluate log 4 9 on your calculator.

Section 6.6: Logarithmic and Exponential Equations Summary We will work with the properties of logarithms and exponents to solve equations.