Functions. EXPLORE \g the Inverse of ao Exponential Function
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1 ifeg Seepe3 Functins Essential questin: What are the characteristics f lgarithmic functins? Recall that if/(x) is a ne-t-ne functin, then the graphs f/(x) and its inverse,/'~\x}, are reflectins f each ther abut the line j/ = x. The dmain f/(x) is the range ff 1(x), and the range f/(x) is the dmain ff~1(x). Vide Tutr EXPLORE \g the Inverse f a Expnential Functin The graph f /(x) = 2X is shwn. Graph r"~1(x) by fllwing these steps. A Cmplete the table by writing the image f each pint n the graph fj[x) after a reflectin acrss the line y = x. Pint n the graph f f(x) (-2,0.25) -*- (0.25, -2) (-1,0.5) -* Pint n the graph's image Name..Class- Date- (0, 1) -»- (1, 2) -> a. u 01 B (2, 4) -* Plt and label the image f each pint n the crdinate plane. C Use the images f the pints t sketch the graph f/~1(x). REFLECT 1a. What are the dmain and range f/ *(x)? I! 1b. Des the graph f/ (x) have any asympttes? Explain hw yu knw. 1c:. Hw d the values f/ (x) change as x increases withut bund? Chapter Lessn 3
2 C 9-t2.F.BF5(+) ENGAGE Defining Lgarithmic Fractins A lgarithm is the expnent t which a base must be raised in rder t btain a given value. Fr example, 23 = 8, s the lgarithm base 2 f 8 is 3, and yu write lg, 8 = 3. Definitin f Lqariti Fr psitive numbers y and b (b ^ 1), the lgarithm f y with base b is written lgfy and is defined as fllws: lgf y = x if and nly ifbx = y. \^_^. _^ This definitin means that every statement abut expnents can be cnverted int an equivalent statement abut lgarithms, and vice versa. Nte that yu read lg^x as "the lgarithm base b f x" r "lg base b fx." A lgarithmic functin with base b is the inverse f the expnential functin with base b. Fr instance, the inverse f/(x) = 2X is/-1(jc) = Ig2 x, the graph f which yu sketched in the Explre. The table describes tw special lgarithms. Name Cmmn lgarithm Natural lgarithm Special Lgarithms Base 10 e Ntatin Write lg x instead f Ig10 x. Write In x instead f lgex. -'' The number e is the value that thex expressin (1 + - ) appraches as x increases withut bund, e is an irratinal number, apprximately a. Explain, in terms f a lgarithmic functin, hw t write 72 = 49 as an equivalent statement invlving a lgarithm. 2b. Explain, in terms f an expnential functin, hw t write lg 1000 = 3 as an equivalent statement invlving an expnent. 2c. The input f/(x) = 2X is an expnent and the utput is a pwer f 2. Describe the input and utput ff~1(x) = Ig2 x. Give a specific example. 2d. Find In ~ by letting In \ x and writing this statement in an equivalent frm that invlves an expnent. Explain yur reasning frm that pint n. Chapter Lessn 3
3 EXAMPLE \g Lgarithmic Factins Find each value f f(x) = Ig2x. A /(16) Write the functin's input as a pwer f 2. The functin's utput is the expnent = 2, s/(16)= B /(64) 64 = 2,s/(64) = C,s/{i)=. 1 = 2,s/g)= /(I) 1 = 2,s/l)= a E 3a. Is it pssible t evaluate/(o)? Why r why nt? 3b. fif(x) - Ig2x, between which tw integers des/(40) lie? Explain. 3c. Estimate g(95) fr g(x) = \gx, withut using a calculatr. Explain. 3d. Withut using a calculatr, explain hw yu knw that In 20 > lg 20. Chapter Lessn 3
4 CC.9-i2.F.IR7e EXAMPLE \g a Lgarithmic Graph f(x) = \g,x. A What is the range f/(x)? (Think: what pwers can yu raise t?). What is the dmain f/(jc)? (Think: what values can be btained by raising t a pwer?) B Cmplete the table f values. C Plt the pints. Cnnect them with a smth curve. f(x) 4a. Hw is the graph ff(x) lgj x related t the graph if(x) Ig2x frm the Explre? 2 Why des this make sense? 4b. What pint d the graphs f/(x) = lgj x and/(x) = Ig2 x have in cmmn? Why? 4c. Describe the end behavir ff(x) = lgj x. 4d. Hw is the graph f/(x) = lgj x related t the graph f/(x) = f ^J? Chapter Lessn 3
5 PRACTICE! ««?*?;*;: JSSfV K-5 1. The graph f/(x) = 3* is shwn. a. Use the labeled pints t help yu draw the graph ff~i(x). Label the crrespnding pints nf~l(x). b. Write the inverse functin, f~\x}, using lgarithmic ntatin. c. State the dmain and range f/ l(x). Find each value f f(x) = Ig4x. 2- /(16) Find each value f f(x) = lgx. 5. /(10,000) 6. /(O.I) Find each value f f(x) = lg^ /(16) 9. /(I) '» Evaluate each expressin. u Ig Ig lg 1,000, Fr/(x) = lg x, between what tw integers des/(6) lie? Explain. 0 I 15. Explain hw yu can estimate the value f In 10 withut using a calculatr. 16. What is the value f lg, b fr b > 0 and b ± 1? Explain. Chapter Lessn 3
6 Graph each lgarithmic functin. - 2 L4J^-4-i~4 a» I J 1 ' ' ' ; i I 19. The graph f what lgarithmic functin is shwn? Explain yur reasning. 20. Name sme values yu wuld chse fr x if yu were pltting pints t sketch the graph f/(x) = lg x withut using a calculatr. Explain. 21. The graph f/(x) = lgfex passes thrugh the pints (1, 0) and (36, 2). What is the value f bl Explain. 22. When yu fld a sheet f paper in half x times, the functin/(x) = 2* gives the number f sectins that are created by the flds. X c! a. Describe the input and utput f the functin/"1^) = Ig2xin the prblem cntext. b. Use/ (x) t find the number f flds needed t create 64 sectins. e. Assume that a sheet f paper has an area f 1. Write an expnential functin g(x) that gives the area f a sectin f the paper after being flded in half xtimes. d. Write the rule fr g :(x) and describe the functin's input and utput. Chapter Lessn 3
7 Name- Class. Date j Additinal Practice/ Write each expnential equatin in lgarithmic frm.. 37 = = = 125 Write each lgarithmic equatin in expnential frm. 4. lgi 100,000 = 5 5. Ig41024 = 5 6. Ig9729 = Evaluate by using mental math. 7. lg 1,000, lg lg Ig Ig Ig5 625 Use the given x-values t graph each functin. Then graph its inverse. Describe the dmain and range f the inverse functin. 13. f(x) = 2*;x = -2, -1,0, 1,2, 3,4 14. f(x) = \x = -3,-2I-1,0, 1,2, 3 i" : 10 j i : i a u 01 e [ j $ 10 t. 3: c [ 4 9 ; 1 3 1P 1? : i r 1,10 X Slve. 15. The hydrgen in cncentratin in mles per liter fr a certain brand f tmat-vegetable juice is a. Write a lgarithmic equatin fr the ph f the juice. b. What is the ph f the juice? Chapter Lessn 3
8 Prblem Slving/ The acidity f rainwater varies frm lcatin t lcatin. Acidity is measured in ph and is given by the functin ph = -lg[h+], where [H+] represents the hydrgen in cncentratin in mles per liter. The table gives the [H*] f rainwater in different lcatins. 1. Find the acidity f rainwater in eastern Ohi. a. Substitute the hydrgen in cncentratin fr rainwater in eastern Ohi in the functin fr ph. Hydrgen In Cncentratin f Rainwater Lcatin [HI (mles per liter) b. Evaluate the functin. What is the acidity f rainwater in eastern Ohi t the nearest tenth f a unit? Central Califrnia Eastern Texas Eastern Ohi Find hw the acidity f rainwater in central Califrnia cmpares t the acidity f rainwater in eastern Ohi. a. Write a functin fr the acidity f rainwater in central Califrnia. b. Evaluate the functin. What is the acidity f rainwater in central Califrnia t the nearest tenth f a unit? c. Cmpare the ph f rainwater in the tw lcatins. Is the ph f rainwater in eastern Ohi greater than r less than that in central Califrnia? By hw much? Chse the letter fr the best answer. 3. What is the ph f rainwater in eastern Texas? A ph = 3.7 C ph = 4.4 B ph=4.0 D ph = What is the ph f a sample f irrigatin water with a hydrgen in cncentratin f 8.3 x 10~7 mles per liter? A ph = 6.1 C ph = 6.3 B ph = 6.2 D ph = Nick makes his wn vegetable juice. It has a hydrgen in cncentratin f 5.9 x 10~6 mles per liter. What is the ph f his vegetable juice? F ph = 4.9 H ph = 5.1 G ph = 5.0 J ph What is the ph f a shamp sample with a hydrgen in cncentratin f 1.7x 10~8 mles per liter? F ph = 7.4 H ph = 7.8 G ph = 7.6 J ph = 8.0 I Chapter Lessn 3
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