5.1 Exponential and Logarithmic Functions

Transcription

1 Math 0 Student Notes. Eponential and Logarithmic Functions Eponential Function: the equation f() = > 0, defines an eponential function for each different constant, called the ase. The independent variale e may assume any real value. The domain of f is the set of all real numers, and the range of f is the set of all positive real numers. we require the ase to e positive to avoid imaginary numers such as (-) / Basic Properties of the graph of f() =, > 0 ) all graphs pass through the point (0,) ) All graphs are continuous with no holes or jumps. ) The -ais is a horizontal asymptote. ) If > then increases as increases. ) If 0 < <, then decreases as increases. 6) The function f is one-to-one Eample: Graph f ( ) = Start y finding some ordered pairs. X - - Y

2 Math 0 Student Notes The natural eponential function e e =.7... Graph is etween and Graph of e and e - e 0 e e ,, 0, 69

3 Math 0 Student Notes Compound Interest If a principal P is invested at an annual rate r compounded n times a year, then the amount A in the account at the end of t years is given y A = P + r n nt A = rt Pe Eample: If you deposit $6000 in an account paying % compounded daily, how much will you have in the account in years? Eample: You invest $000 at an annual rate of %. Find the alance after 7 years if interest is compounded continuously. 70

4 Math 0 Student Notes. Logarithmic Functions Logarithmic functions are inverses of eponential functions. f : = y If f : y = then Definition of Logarithmic Function For > 0 and, y y = log = is equivalent to The log to the ase of is the eponent to which must e raised to otain. The log function f() log a has domain (0, ) and range (-, ) log( ) 0 0 Logarithmic-Eponential Conversions a) ) log = log = / Eponential Logarithmic conversions a) 9 = 7 ) = 9 7

5 Math 0 Student Notes Solutions of the Equation a) y = log y = log ) log = Rule of Thum the ase of the log is the ase of the eponent The value of the log is the eponent Properties of Logarithmic Functions ) log = 0 ) log = ) log = log ) = Common Logartihms log = log 0 Measuring Loudness (Intensity) The loudness of a sound is measured using deciels. The faintest sound perceptile to most individuals is assigned and intensity of I o, and a sound of intensity I has a deciel rating of D 0log I = I0 Eample: The sound of a lue whale can reach an intensity of 6. X 0 times that of I 0. find the deciel rating. 7

6 Math 0 Student Notes Common Decial Ratings Near total silence - 0 db A whisper - db Normal conversation - 60 db A lawnmower - 90 db A car horn - 0 db A rock concert or a jet engine - 0 db Natural Logarithms ln = log0 The eponential function and the natural log are inverse functions. ln = y y e = Properties of natural Logarithms. ln = 0 ecause 0 e =. ln e = ecause e = e. ln e = ecause ln e =. if ln = ln y then = y 7

7 Math 0 Student Notes. Laws of Logarithms ) log MN = log M + log N ) log M = log N M log ) log p M = plog M N Eample: log a+ log / log c y Eample: log z = Change of Base Let a and e positive real numers with a and. Then for any positive real numer u, log u loga u = log a Eample: Convert log6 to ase 0 = Convert log6 to ase e = **Common Errors log M. log M log log N ( ) p. ( log ). log M + N log M + log N M plog M N 7

8 Math 0 Student Notes. Solving Eponential and Logarithmic Equations Eample: solve = Eample: Solve ( ) log + + log = Eample: Solve lo g = / log7 + log log Eample: Solve - = 7

9 Math 0 Student Notes Eample: Solve e 7e + Compound Interest r A = P + n nt Eample: How many years will it take the money to doule if it is invested at 6% compounded daily? 76

10 Math 0 Student Notes. Modeling with Eponential and Logarithmic Functions Growth and Decay Applications Eample: The world population is given y the following formula 0.069t P = 6e where P is in millions and t = 6, represents 006 Which year will the population reach 7. illion? Eample: R = The ratio of caron to caron is given y the following: t / e 0 If a newly discovered fossil has R =, estimate the age of the fossil 9 77

11 Math 0 Student Notes Newton s Law of cooling You uy a 6 pack of eer on a hot summer day with a temp of 90 o. The eer is placed in a refrigerator with a constant temp of 0 o. If the eer cools to 60 o in one hour, when will the eer e at the perfect drinking temp of o? kt T t = T + D e Where () s 0 D0 is the initial temperature difference, T s is the surrounding temperature 7