Logarithms and Logarithmic Functions

Logarithms and Logarithmic Functions Essential Question What are some of the characteristics of the graph of а logarithmic function? Every exponential function of the form f(x) = ь х , where Ь is а positive real number other than 1, has an inverse function that you can denote Ьу g( х) = logь х . This inverse function is called а logarithmic function with base Ь. EXPLORATION: Rewriting Exponential Equations Work with а partner. Find the value of х in each exponential equation. Explain your reasoning. Then use the value of х to rewrite the exponential equation in its equivalent logarithrn.ic� l�f>'ci}1. i.\ / -;fJ\/t, а. 8 2х = �1, .-=- 1(' �! d. 5 х = 1 х Ь. 3 = 9 31 � -;_ ;.. ( х е. 5 х = 2 1 5 f. g J... 5 x 4 Notetaking with Vocabulary ln your own words, write the meaning of each vocabulary term. common logarithm logarithm ofJfi.With base -h1function natural logarithm Core Concepts Definition of Logarithm with Base Ь Let Ь and у Ье positive real numbers with Ь logь у and is defined as logь у == х * 1. The logarithm ofу with base Ь is denoted Ьу <'iraud oulx The expression logь Y is read as "log base Ь ofy." u:;> х ь = у. �\ Parent Graphs for Logarithmic Functions Yl\�rf Thegraph of f(x) = logь х isshown below fo Ь > 1 1d fo О < Ь < 1. Bec ause f(x) = logьx and g(x) = ьх a re inverse fun cti ons, thegraph of f(x) = logьx isthe reflecti on ofthe graph of g(x) = b·t in t\1e Jine у = х. Graph of /(х) = logь x for Ь > G1·apl1 of f(x) = logь x for О < Ь < 1 / / / х ,1 ,1 f(x) ,1 ,1 Notethatthey-a�a;erti c al asymptote ofthegraph of f(x) = logь�) = logьx, Thedom a in of f(x) = logьxisx > О, and the range is all realnumbers. Notes: Extra Practice ln Exercises 1-4, rewrite the equation in exponential form. 1. log1;o (000 -- - = l,o�; 3. log101 = О 3 ь (О : l , 2. 1 log525 4. log114 64 = -3 10 = -2 - -z� (Ь у\.ь ч Notetaking with Vocabulary (continued) ln Exercises 5-8, rewrite the equation in logarithmic form. 5. = 144 ;}k�� - lo�, 7. 6. ..... 216 113 6 20- 1 J_ = 20 !в�;?." CQ! 8. 4° = 1 10. log 1;8 1 ln Exercises 9-12, evaluate the logarithm. 9. log4 64 )j;-)/; ::.. х 1,,( :: "ч 11. - LI � 12. r, \ ;,� � ;��� 168 Algebra 2 Student Journal 5 - -· l 32,ь I 1 log1/2s- l 2 15,-- Х;..С> \о�� ">< � 2·l·'2 ,х ..... __.. \ � (оЧ d 1 log232 c; fг( ( ( а Jc(,t I l \. \' t !r ) ( -- /;� Jt ,, <(-.' Copyright © Big ldeas Learning, LLC AII rights reseNed. ln Exercises 13 and 14, slmplify the expression. ln Exercises 15 and 16, find the inverse of the function. у = 1 '-- t о 15. х + 10 --t () 16 +�-- _J__$f у J е.у /� ,�( у-10') � Х \о�(у-10) 15 - \ .У = ln(2x) - 8 � - 2'!<-_ � _ --:,г... -=-- '7-- _ --� � }·---- ln Exercises 1 �d 18, graph the function. Determine tfie asymptote of the function. 18.У / J · т . . .. . . log 112 .x - 4 f--+----+-.,+---+--------+-----+---1- "'"" _,,,•.,. EXPLORATION: Graphing Exponential and Logarithmic Functions Go to ВigbleasMatli.com for an interactive tool to investigate this exploration. Work with а partner. Complete each tаЫе for the give11 expoнential fu11ctio11. Use the results to complete the tаЫе for the giveп logarithmic fuпctioп. Explaiп your reasoпiпg. Тhеп sketcl1 tl1e graphs offапd g iп tl1e same coordiпate р!апе. а. 1 :(х) = 2' 1 1 : иs -2 о -1 2 vl 1 rz_ -2 -1 о 2 l-l--,----l----1--4---+- �-1--'- - г- >- - --------------.,.....; 1--+-!-· -1-- ...___ ь. -2 -1 о 1--- -•-r-- - ..,.___._...,'--'___.,__,___,_--"'- 2 1:(х) = 1о х 1:и = IOQ10 Х -2 -1 о 2 f----.f-- -1--- - -·1---r--- - -- ---- -1- - �- -1- ---- EXPLORATION: Graphing Exponential and Logarithmic Functions Go to BigldeasMat/1.com for an interactive tool to investigate tliis exploration. Work ,vith а partner. Complete еасЬ tаЫе for the given expone11tial ftшctio11. Use tЬе results to complete tЬе tаЫе for tЬе given logaritЬmic ftшction. Explain your reasoпiпg. Тhеп sketch tl1e grapl1s ofj апd g iп tЬе same coordiпate рlапе. а. -2 о -1 -2 2 -1 о 2 ,-.. ,1--,1--,--+--+--+--+----+-·""' �"... -.1--- --- ---- +--+----<е-+-- -- ь. 1 : (х ) 1 :и = -2 -1 -2 -1 1о х = 10910 х 1 о 2 2 о - � ,- - ---- - --,__, -