Page of 7 Eponential and Logarithmic Functions Eponential Functions and Their Graphs: Section Objectives: Students will know how to recognize, graph, and evaluate eponential functions. The eponential function f() with base b is denoted by f( ) = b where b > 0, b, and is any real number. The eponential function is called a transcendental function. Graphs of Eponential Functions For the equation f () = b. The domain is (, ), so its continuous for all real numbers 2. The range is (0, ). The y-intercept is (0, ) 4. y = 0 is a horizontal asymptote (only on one end) 5. f () is increasing if b > 6. f () is decreasing if 0 < b < 7. f () is one to one E: Graph the following eponential functions. a. f() = 2 b. f() = 2 - = (/2) E: Graph each of the following on the same coordinate aes. a. f() = b. g() = + c. h() = 2 d. k() = - E: See what happens if we change the base. a. g() = 4 b. f() = 4 - + 2 c. h() = 4 - + 2 Eponential Functions Properties. Eponent laws: y ( ) ( ) a a = a a = a ab = ab y + y y a a a = a = b b b 2. a = a y iff = y. For 0, a = b iff a = b y
Page 2 of 7 The Natural Base e e is an irrational number, where e = 2.7828828459 The eponential function with base e is called the natural eponential function f( ) = e It can be shown that f () = ( + /) e as. [ + ] 2 2 2.25 4 2.444 2 2.604 65 2.7457 8,760 2.78 525,600 2.7828 E: Draw the graph of f() = e -2 Applications Let A be the amount in the account after n pay periods, let P be the principal, and let be the periodic interest rate. Then the compounded interest after one year can be calculated by using, A = P( + ) n The compound interest formula after t years is A = P( + r/n) nt A is the amount in the account after t years. P is the principal. r is the annual interest rate. n is the number of pay periods per year. An investment of $5,000 is made into an account that pays 6% annually for 0 years. Find the amount in the account if the interest is compounded: a. annually (n = ) ($8954.24) b. quarterly (n = 4) ($9070.09) c. monthly (n = 2) ($9096.98) d. daily (n = 65) ($90.4) As n increases, so does A, but the rate of increase slows. What would happen if n?
Working with A = P( + r/n) nt let n/r = or r/n = / we get A= P + then as n, also goes to, and the following continuously compounded interest formula can be derived. A = Pe rt A is the amount in the account after t years. P is the principal. r is the annual interest rate. rt Page of 7 E: Continue the eample previous with the interest compounded continuously. Logarithmic Functions: Section Objectives: Students will know how to recognize, graph, and evaluate logarithmic functions, rewrite logarithmic functions with a different base, use properties of logarithms to evaluate, rewrite, epand, or condense logarithmic epressions. Since the eponential function f() = b is one-to-one, its inverse is a function. The function given by f( ) = log b where > 0, b > 0, and b is called the logarithmic function with base b. Conversely, the logarithmic function with base b is the inverse of the eponential function with base b; thus y = log b if and only if = b y E: Evaluate each of the following. a. f() = log 2 2 b. f() = log c. f() = log 4 2 d. f() = log 0 (/00) (the two statements are equivalent)
Page 4 of 7 We call the logarithmic function with base 0 the common logarithmic function. This is the function that corresponds to the LOG button on our calculators. The common logarithmic function is the one function for which we do not write the base. Graphs of Logarithmic Functions The graph of the logarithmic function comes directly from the properties of the graph of the eponential function and the inverse relationship. Inverse functions graphs are symmetric around the y = line. f( ) = log b f( ) = 2 and g( ) = log Look at 2 For The domain is (0, ). The range is (, ). The -intercept is (, 0). The y-ais is a vertical asymptote. The function is increasing (b > ) and decreasing (0 < b < ) Continuous over its entire domain E: Sketch the graph of the following. a) y = log 0 b) y = log 0 ( + 2) c) y = log 0 ( + 2) The Natural Logarithmic Function The logarithmic function with base e is called the natural logarithmic function and is denoted by: f( ) = log = ln, > 0 f( ) = ln is the inverse of g( ) = e Draw the graph of f() = ln E: Find the domain of the following and graph a. f( ) = ln( 2) b. g( ) = ln( 2 ) Properties of Log functions Properties of Natural Logs. log b = 0 because b 0 =. ln = 0 because e 0 = 2. log b b = because b = b 2. lne = because e =e a =. log b b log = and a. lne = and 4. If log b = log b y, then = y. 4. If ln = ln y, then = y You can see that these are the same since ln is just a log base e e ln e =
E: Simplify a. 4= 4 b. 5 5 = 6 c. 6 log 20 = Page 5 of 7 E: Simplify a. ln e b. ln5 e c. ln More properties of Logarithms d. 2lne Since y = log b if and only if = b y, then the following properties of logarithms are similar to the eponent properties. Let b be a positive real number such that b, let n be a real number, and let u and v be positive real numbers. log b (uv) = log b u + log b v log b (u/v) = log b u log b v log b u n = n log b u ln(uv) = lnu + lnv ln(u/v) = lnu lnv ln u n = n ln u The use of these properties is usually known as the epansion of logs E: Epand the logarithmic epression. Write each log as ln 2 or ln a) ln 6 b) ln(2/27) E: Verify that log0 = log0 00 00 E: Epand Logarithms completely a. log 5 y 4 b. ln E: Rewrite as a single logarithm a. log log( ) 2 5 7 + + b. [ log + log ( + ) ] 2 2
Page 6 of 7 Change of Base Our calculators have only two buttons for logarithmic functions, base 0 and base e. If we want to evaluate logs with other bases we need this formula. The Change-of-Base Formula Let a, b, and be positive real numbers such that a and b. Base b Base 0 Base e logb log0 ln loga= loga= loga= log a log a lna Use calculator to solve log 0 4 b Eponential and Logarithmic Equations: Section Objectives: Students will know how to solve eponential and logarithmic equations. 2 Ways To Solve An Eponential or Log Equation. Use one to one property 2. Use inverse property 0 One-to-One Properties. a = a y if and only if = y. 2. log a = log a y if and only if = y. Inverse Properties. log a a = log 2. a a = E: Solve a. 2 = 2 b. ln ln= 0 c. d. e = 7 e. = 9 ln= f. log0= Strategies for Solving Eponential and Log Equations. Isolate the eponential or Log 2. Rewrite the original equation in a form that allows you to use the one-to-one property.. Rewrite the eponential equation in Log form and apply the Inverse Property 4. Rewrite Log equation in eponential form and apply the Inverse Property. Solve: a. 4 = 72 b. ( 2 ) = 42 c. e + 5= 60 d. 2 5 2( t ) 4=
Solving Eponentials of Quadratic Type: 2 Solve: e e + 2= 0 Eponentiating Both Sides of the Equation Solve: ln = ln = 2 log (5-)=log (+7) 5 + 2ln = 4 2 log 5 = 4 Etraneous Solutions log 5+ log ( ) = 2 0 0 Check the answers!!! Applications How long would it take for an investment of $500 to double if the interest were compounded continuously at 6.75%? Page 7 of 7