4.3 COLLEGE ALGEBRA. Logarithms. Logarithms. Logarithms 11/5/2015. Logarithmic Functions

Transcription

1 0 TH EDITION COLLEGE ALGEBRA 4. Logarithic Fuctios Logarithic Equatios Logarithic Fuctios Properties of LIAL HORNSBY SCHNEIDER The previous sectio dealt with epoetial fuctios of the for y = a for all positive values of a, where a. The horizotal lie test shows that epoetial fuctios are oe-to-oe, ad thus have iverse fuctios. The equatio defiig the iverse of a fuctio is foud by iterchagig ad y i the equatio that defies the fuctio. Startig with y = a ad iterchagig ad y yields y a a Here y is the epoet to which a ust be raised i order to obtai. We call this epoet a logarith, sybolized by log. The epressio log a represets the logarith i this discussio. The uber a is called the base of the logarith, ad is called the arguet of the epressio. It is read logarith with base a of, or logarith of with base a. y Logarith For all real ubers y ad all positive ubers a ad, where a, y y log a if ad oly if a. A logarith is a epoet. The epressio log a represets the epoet to which the base a ust be raised i order to obtai

2 Epoet Logarithic for: y = log a Base Epoet Epoetial for: a y = Base Eaple SOLVING LOGARITHMIC EQUATIONS Solve a. log 7 log 7 7 Write i epoetial for. 7 Take cube roots Eaple Check: SOLVING LOGARITHMIC EQUATIONS log 7 log Origial equatio? Let = ⅔.? Write i epoetial for True The solutio set is Eaple Solve b. log4 The solutio set is {}. SOLVING LOGARITHMIC EQUATIONS log4 4 (4 ) Write i epoetial for. a ( a ) 4 ( ) 4. - c. Eaple Solve log 7 49 The solutio set is. 6 SOLVING LOGARITHMIC EQUATIONS 49 7 (7 ) 7 Write i epoetial for. Write with the sae base. 7 7 Divide by. 6 Power rule for epoets. Set epoets equal. 4. -

3 Logarithic Fuctio Logarithic Fuctio If a > 0, a, ad > 0, the f ( ) log a defies the logarithic fuctio with base a. Epoetial ad logarithic fuctios are iverses of each other. The graph of y = is show i red. The graph of its iverse is foud by reflectig the graph across the lie y = Logarithic Fuctio Logarithic Fuctio The graph of the iverse fuctio, defied by y = log, show i blue, has the y-ais as a vertical asyptote Sice the doai of a epoetial fuctio is the set of all real ubers, the rage of a logarithic fuctio also will be the set of all real ubers. I the sae way, both the rage of a epoetial fuctio ad the doai of a logarithic fuctio are the set of all positive real ubers, so logariths ca be foud for positive ubers oly LOGARITHMIC FUNCTION f ( ) log a Doai: (0, ) Rage: (, ) For () = log : () ¼ ½ 0 4 () = log a, a >, is icreasig ad cotiuous o its etire doai, (0, ). LOGARITHMIC FUNCTION f ( ) log a Doai: (0, ) Rage: (, ) For () = log : () ¼ ½ 0 4 The y-ais is a vertical asyptote as 0 fro the right

4 LOGARITHMIC FUNCTION f ( ) log a Doai: (0, ) Rage: (, ) For () = log : () ¼ ½ 0 4 The graph passes through the poits,,,0, ad a,. a LOGARITHMIC FUNCTION f ( ) log a Doai: (0, ) Rage: (, ) For () = log / : () ¼ ½ 0 4 () = log a, 0 < a <, is decreasig ad cotiuous o its etire doai, (0, ) LOGARITHMIC FUNCTION f ( ) log a Doai: (0, ) Rage: (, ) For () = log / : () ¼ ½ 0 4 The y-ais is a vertical asyptote as 0 fro the right. LOGARITHMIC FUNCTION f ( ) log a Doai: (0, ) Rage: (, ) For () = log / : () ¼ ½ 0 4 The graph passes through the poits,,,0, ad a,. a Characteristics of the Graph of f ( ) log a. The poits,,,0, ad a, are o the a graph.. If a >, the is a icreasig fuctio; if 0 < a <, the is a decreasig fuctio.. The y-ais is a vertical asyptote. 4. The doai is (0,), ad the rage is (, ). Cautio If you write a logarithic fuctio i epoetial for, choosig y- values to calculate -values, be careful to write the values i the ordered pairs i the correct order

5 Properties of Sice a logarithic stateet ca be writte as a epoetial stateet, it is ot surprisig that the properties of logariths are based o the properties of epoets. The properties of logariths allow us to chage the for of logarithic stateets so that products ca be coverted to sus, quotiets ca be coverted to differeces, ad powers ca be coverted to products. Properties of For > 0, y > 0, a > 0, a, ad ay real uber r: Property Product Property log y log log y a a a Descriptio The logarith of the product of two ubers is equal to the su of the logariths of the ubers Properties of For > 0, y > 0, a > 0, a, ad ay real uber r: Property Quotiet Property loga loga logay y Descriptio The logarith of the quotiet of two ubers is equal to the differece betwee the logariths of the ubers. Properties of For > 0, y > 0, a > 0, a, ad ay real uber r: Property Power Property loga loga logay y Descriptio The logarith of a uber raised to a power is equal to the epoet ultiplied by the logarith of the uber Properties of Two additioal properties of logariths follow directly fro the defiitio of log a sice a 0 = ad a = a. log 0 ad log a a a Eaple 4 Rewrite each epressio. Assue all variables represet positive real ubers, with a ad b. a. log 6(7 9) log ( 7 9) log 7log 9 Product property

6 Eaple 4 Rewrite each epressio. Assue all variables represet positive real ubers, with a ad b. b. log9 7 log9 log9 log97 7 Quotiet property Eaple 4 Rewrite each epressio. Assue all variables represet positive real ubers, with a ad b. c. log log log ( ) log Power property Eaple 4 Rewrite each epressio. Assue all variables represet positive real ubers, with a ad b. q d. log a 4 Use paretheses pt to avoid errors. q 4 loga log log log ( log log ) 4 a a aq ap a t pt log log log q ( log p 4 log t) a a a a a log log log q log p 4 log t a a a a a Be careful with sigs e. Eaple 4 Rewrite each epressio. Assue all variables represet positive real ubers, with a ad b. log a log log log Power property a a a f. Eaple 4 Rewrite each epressio. Assue all variables represet positive real ubers, with a ad b. log b y z y logb z y logb z log log log b by bz a a Power property Product ad quotiet properties f. Eaple 4 Rewrite each epressio. Assue all variables represet positive real ubers, with a ad b. log b y z y logb z log b log by log bz logb logby logbz Power property Distributive property

7 Eaple Write the epressio as a sigle logarith with coefficiet. Assue all variables represet positive real ubers, with a ad b. a. log ( ) log log log ( ) log log log Product ad quotiet properties ( ) Eaple Write the epressio as a sigle logarith with coefficiet. Assue all variables represet positive real ubers, with a ad b. b. log log a a log log log log a a a a Power property log a Quotiet property Eaple Write the epressio as a sigle logarith with coefficiet. Assue all variables represet positive real ubers, with a ad b. c. logb logb logb logb logb logb log log ( ) log b b b Power properties Eaple Write the epressio as a sigle logarith with coefficiet. Assue all variables represet positive real ubers, with a ad b. c. logb logb logb ( ) Product ad quotiet log b properties log b Rules for epoets Eaple Write the epressio as a sigle logarith with coefficiet. Assue all variables represet positive real ubers, with a ad b. c. logb logb logb log Rules for epoets b log Defiitio of a / b Cautio There is o property of logariths to rewrite a logarith of a su or differece. That is why, i Eaple (a), log ( + ) was ot writte as log + log. Reeber, log + log = log ( ). The distributive property does ot apply i a situatio like this because log ( + y) is oe ter; log is a fuctio ae, ot a factor

8 Eaple 6 WITH NUMERICAL VALUES Assue that log 0 =.00. Fid each logarith. log 4 a. 0 log 4 log log (.00) b. log0 0 log0 log0 log0 0 log Theore o Iverses For a > 0, a : loga a aa ad log Theore o Iverses By the results of this theore, log , log, ad log r k r k. The secod stateet i the theore will be useful i Sectios 4. ad 4.6 whe we solve other logarithic ad epoetial equatios