Unit 1 Exponentials and Logarithms

Transcription

1 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 1 Unit 1 Eponentils nd Logrithms (2) Eponentil Functions (3) The number e (4) Logrithms (5) Specil Logrithms (7) Chnge of Bse Formul (8) Logrithmic Functions (10) Lws of Logrithms (12) Solving Eponentil & Logrithmic Equtions Know the menings nd uses of these terms: Eponentil epression, eponentil function Logrithmic epression, logrithmic function Bse of n eponentil or logrithmic epression Eponent of n eponentil or logrithmic epression Common logrithm Nturl Logrithm Etrneous solution Review the menings nd uses of these terms: Asymptote

2 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 2 Eponentil Functions Definition: The eponentil function with bse, such tht is positive rel number other thn 1, is defined by f, 0. Emples: f 2 Domin:, Rnge: 0, Key Point: (0, 1) Asymptote: y = 0 Recll tht n symptote is line tht grph pproches s vlues grow without bound. f 4 5 If the bse > 1, the function will increse over its domin with symptotic behvior s. If the bse < 1, the function will decrese over its domin with symptotic behvior s. A specil number frequently ssocited with eponentil functions is e.

3 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 3 The number e Definition: The nturl eponentil bse is the number e, such tht e n 1 lim n n. Identify the bse of the eponentil function nd use trnsformtions to determine the domin, rnge, nd symptote of ech given function. E. 1: 1 f( ) 4 2 An eponentil function with bse of e is clled nturl eponentil function: f e. E. 2: g( ) 2e 3

4 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 4 Logrithms The inverse of n eponentil epression is logrithm. Definition: Let be positive rel number other thn 1. Then the logrithm with bse, denoted s log, is defined s y follows: log. y All logrithmic epressions stisfy one of the following properties: log 1 0, log 1, log Evlute. E. 1: log 6 1 n n E. 2: log 8 8 More specific to bove, the inverse of n eponentil epression with bse is logrithm with bse. E. 3: log 7 49 Thus the vlue of logrithmic epression is equl to the eponent of bse which equls the input.

5 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 5 Rewrite ech sttement into its inverse form. Specil Logrithms E. 1: 3 4 = 81 E. 2: 7 2 = 49 E. 3: 5 = 20 E. 4: log 6 36 = 2 Two specil logrithms cn be written without bse. Definition: The common logrithm is the logrithm with bse 10 such tht log 10 = log. Definition: The nturl logrithm is the logrithm with bse e such tht log e = ln. E. 5: log 2 64 = 6 E. 6: log 11 = 2 (Note: In some higher levels of mthemtics, log my ctully refer to nturl logrithm.)

6 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 6 Evlute. Approimte s necessry to five digits. Rewrite ech sttement into its inverse form. E. 1: log 1000 E. 1: 10 = 500 E. 2: log 0.01 E. 2: e = 20 E. 3: ln e A clcultor my be helpful in pproimting some logrithmic epressions: E. 3: log = 3 E. 4: ln = 5 E. 4: log 400 E. 5: ln 40

7 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 7 Chnge of Bse Formul Evlute. Approimte s necessry to five digits. Mny logrithms my not hve n obvious eponentil vlue. Common nd nturl logrithms cn esily be used to pproimte eponentil vlues but second lyer pproch is necessry with other bses. E. 1: log log m log b m ln m log m log ln log b E. 2: log E. 3: log

8 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 8 Logrithmic Functions Definition: The logrithmic function with bse, such tht is positive rel number other thn 1, is defined by f log, 0. Domin: 0, Rnge:, Key Point: (1, 0) Asymptote: = 0 Observe tht the definition of logrithmic function, in conjunction with the third property of logrithms, stisfies the Property of Inverse Functions. Tht is, logrithmic function of bse is the inverse function to n eponentil function of bse. If the bse > 1, the function will increse over its domin with symptotic behvior s 0. If the bse < 1, the function will decrese over its domin with symptotic behvior s 0. Wheres the symptote of n eponentil function is horizontl, the symptote of logrithmic function is verticl. Further, the logrithmic function hs n -intercept insted of y-intercept like the logrithmic function. To sketch the grph of logrithmic function not bse 10 or e, use the chnge of bse formul.

9 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 9 Emple: f 2 ln ln 2 Identify the bse of the eponentil function nd use trnsformtions to determine the domin, rnge, nd symptote of ech given function. E. 1: f 5 ( ) 2log ( 1) Furthermore, since logrithmic functions nd eponentil functions re inverses, they stisfy the grphicl properties of inverse functions s illustrted t right with the nturl eponentil function nd the nturl logrithmic function. E. 2: g( ) 2 log 3

10 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 10 Lws of Logrithms The lws of logrithms llow us to rewrite logrithmic epressions so they re esier to mnipulte. Ech lw of logrithms hs n nlogue with the lws of eponents. Rewrite ech logrithm into sum or difference of logrithms so tht no logrithm consists of product, quotient, or power (where possible). E. 1: lo g y Nme Lw of Logrithms Lw of Eponents Product-to-Sum log y log log y Quotient-to-Difference m n m n E. 2: lo g log log log y y m n m n Power-to-Product log n n log m n mn

11 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 11 Rewrite ech logrithm into sum or difference of logrithms so tht no logrithm consists of product, quotient, or power (where possible). Rewrite ech s single logrithm. Simplify where possible. E. 3: log 3 yz 5 2 E. 1: 2log 4 log y log z E. 2: log 2 log y 5log z E. 4: log E. 3: 2ln 3 4 ln2 E. 4: 3ln 5 2ln 6

12 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 12 Solving Eponentil Equtions Procedure for most eponentil equtions: 1. Isolte n eponentil epression on one side. 2. Tke the nturl logrithm (or common logrithm) of both sides. 3. Use the lws of logrithms to rewrite the eponentil epression so tht no vrible remins in the eponent. 4. Apply bsic lgebric nd rithmetic mnipultion to solve for. 5. Use the lws of logrithms to rewrite the solution s pproprite nd pproimte the solution. 6. Check your solution. Some eponentil equtions my use other procedures for prt or ll of the entire process; for emple: qudrtic-type equtions, et.l. Solving Logrithmic Equtions Procedure for logrithmic equtions: 1. Use the lws of logrithms to combine logrithms on ech side s necessry. 2. Apply net step bsed on the structure:. If only one side of the eqution hs logrithm, rewrite the eqution into eponentil form. b. If both sides of the eqution hve logrithms nd the bse is the sme, set the rguments equl to ech other. 3. Apply bsic lgebric nd rithmetic mnipultion to solve for. 4. Check your solutions. Like rdicl equtions, logrithmic equtions cn (nd often do) produce etrneous solutions.

13 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 13 Solve the eqution. Write the solution s single logrithmic epression nd pproimte the solution to five plces. Solve the eqution. Write the solution s single logrithmic epression nd pproimte the solution to five plces. E. 1: E. 2:

14 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 14 Solve the eqution. Write the solution s single logrithmic epression nd pproimte the solution to five plces. Solve ech eqution. E. 4: E. 3: 1 3e 60 E. 5:

15 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 15 Solve the eqution. Write the solution s single logrithmic epression nd pproimte the solution to five plces. E. 6:

16 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 16 Solve the eqution. Write the solution s single logrithmic epression nd pproimte the solution to five plces. E. 7:

17 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 17 Solve the eqution. Write the solution s single logrithmic epression nd pproimte the solution to five plces. Solve the eqution. Write the solution s single logrithmic epression nd pproimte the solution to five plces. E. 8: e 2 3e 2 0 E. 9: 2 4e e 0

18 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 18 Solve the eqution. Check for etrneous solutions. Solve the eqution. Check for etrneous solutions. E. 1: log 2 log 1 log E. 2: ln 3 ln 11 ln4 ln 4

19 HARTFIELD PRECALCULUS UNIT 1 NOTES PAGE 19 Solve the eqution. Check for etrneous solutions. Solve the eqution. Check for etrneous solutions. E. 3: log 6 log E. 4: log 8 log