ࡱ> HJGl_ ybjbjPP 4|y:<\y:<\ t t 88D|<.6:H<J<J<J<J<J<J<>@J<EJ<<N N N H<N H<N N  9,;D,f94<<0<9:A^A4,;,;A@;vQTN DJ<J<^<At > : Activity: Analyzing the Relationship between a Logarithmic Function and Its Inverse This activity will be done after studying algebraic functions, their inverses and basic properties of exponential and logarithmic functions. PART 1: INVESTIGATING A LOGARITHMIC FUNCTION Consider the following function: f(x) = log(x) Make a graph of f(x) using your graphing calculator. The table below lists a series of nine output values from f(x). Fill in the table by using the graph and the TABLE to estimate the missing x values. TABLE 1 XF(x).05.33.891.21.51.752.12.32.5 Using a table of y-values to find the corresponding x-value:  Put  EMBED Equation.3  into Y1 In TBLSET, let TblStart = 0 and "Tbl = 10. As you can see, 1.75 is the y-value somewhere between x = 50 and x = 60. Now, let TblStart = 55 and "Tbl = 1. You can see that 1.75 is a y-value between x = 56 and x = 57. Let TblStart = 56 and "Tbl = 0.1. Once you can be accurate to the tenths place, choose the x-value. Create two lists L1 and L2 in the calculator to store the x and f(x) values. Put the x values in L1 and the f(x) values in L2. Generate a scatter plot of (x, f(x)) using the STAT PLOT feature and the two lists, L1 and L2. Do your x values appear to accurately represent f(x)? If not, modify the values so that they better represent the curve. Generate a new STAT PLOT using L2 and L1 instead of L1 and L2. Hypothesize at least two different families of functions to which this new curve might belong: family 1: __________________________ family 2: __________________________ Using L2 and L1, test your hypothesis with the different regression analysis options and record your results, including the regression equations and R2 values, below. Which of the above selections do you think represents the best fit? Why? If you selected Exponential, you were CORRECT! Congratulate yourself. Look at the exponential equation that you generated. Using your mathematical knowledge and some rounding, estimate g(x), the actual equation of the inverse of f(x): PART 2: Investigating the Inverse Consider the function h(x) = x3. Solve for x if h(x) = 8. What is i(x), the inverse of h(x)? Will i(x) give all the solutions to h(x) = y for any value of y? Explain your answer. Consider the function h(x) = x2. Solve for x if h(x) = 9. Consider the domain of h(x). What would the range of the inverse be? Is there a function, i(x), the inverse of h(x)? Notice that  EMBED Equation.3 will not give all the solutions to h(x)=9. Explain why  EMBED Equation.3  is not the inverse of h(x). Think about the relationship between a function and its inverse. Use that relationship to solve the following: Given: f(x) = log(x). If f(x) = 1.2, find x. Given: g(x) = 10x. If g(x) = 37, find x. part 3: An Application of Logarithms: Hearing and Decibels The decibel is the unit used to measure the intensity of a sound. The decibel is named after Alexander Graham Bell who did much work in the area of sound and loudness. Bell discovered that to obtain a sound that seemed twice as loud as another sound, the intensity (how much sound energy per unit area per second hits the eardrum) of the sound must be multiplied by 10. We call this apparent loudness the "intensity level." The following equation is used to compute the intensity of sound: dB = 10 log (I / Io) where dB represents decibels, I is the intensity of the sound in question (measured in joules per second per square meter), and Io (read as "I naught" ) is the softest sound the human ear can distinguish, being 10-12 joules per second per square meter. Note that "joules per second per square meter" can be expressed more simply as "Watts per square meter" and is written as "W/m2." What exactly does "10-12 joules per second per square meter" (or, more simply, 10-12 W/m2) mean? This intensity corresponds to a sound which will displace particles of air by a mere one-billionth of a centimeter. The human ear can detect such a sound! This faintest sound which the human ear can detect is known as the "threshold of hearing" (TOH). The most intense sound which the ear can safely detect without suffering any physical damage is more than one billion times more intense than the threshold of hearing! It is for this reason that a logarithmic equation is used to measure the intensity of sound. Logarithmic scales are often used when a range of values is extremely large, which is true in the case of the range of intensities that the human ear can detect. Consider this: the human ear can detect a sound as soft as a whisper to as loud as a jet engine flying overhead. That's a big difference in intensity! (Source: Sonic Booms and Logarithms, Robin A. Ward, California Polytechnic State University-San Luis Obispo) Exercises: Complete the worksheet below. Decibels and Logarithms Names:_______________________________________________ Directions: Complete the table using your knowledge of logarithms and the formula:  EMBED Equation.3 , where I = Intensity Sound SourceIntensity (W/m2)Intensity Level (dB)# Times greater than TOHThreshold of hearing (TOH) EMBED Equation.3 Rustling leaves EMBED Equation.3 10Whisper EMBED Equation.3 20Normal conversation60Busy street traffic70Vacuum cleaner80Hearing damage possible EMBED Equation.3 85Lawn mower EMBED Equation.3 Front row at a concert EMBED Equation.3 110Thunderclap (near) EMBED Equation.3 Threshold of pain EMBED Equation.3 130Jet take-off EMBED Equation.3 140Shotgun EMBED Equation.3 Instant perforation of ear drum EMBED Equation.3 160 (Source: Sonic Booms and Logarithms, Robin A. Ward, California Polytechnic State University-San Luis Obispo) Summarize the concepts that were covered in this activity in your own words. Include a discussion about the inverse functions in Parts 1 and 2, and the reading and exercises involving decibels. Put a star next to any concept that was new to you. Extension: Given the formula  EMBED Equation.3  , where [H+ ]is the concentration (in moles per liter) of hydrogen ions: Cindy the saboteur wants to make the pool close so that she wont have to go to swimming practice tomorrow morning. For the pool to remain open, the pH level must be between 7.2 and 7.8. After adding her concoction to the pool, Cindy was able to get a concentration of hydrogen ions of 7.5 10-8 moles per liter. Did she succeed in making the pool unsafe enough to close? Given the formula for Richter Scale,  EMBED Equation.3 , where A is the measured intensity of the earthquake, A0 is the reference intensity, and M is the Richter Scale reading, find the following: Nancy is a scientist who measured the intensity of an earthquake to be 121,000 times the reference intensity. If Nancy needs to report a Richter scale reading to Daniel, a newspaper reporter, what number should Nancy tell Daniel?   4 8 U Y $ 4 L T " , . T V X Z h j l n Z \ ^ ( * ;A_`imy{hflh 6 h H*j;h Uj h UhU\j h UhU\h H*jhU\h EHU#j*;H hU\h CJUVaJjh Ujfh Uh h 5 h 6]h=vh 5 h 5h 2UV D E z { " $ 4 8 : < > @ B D F H J L Ff $IfgdU\ $^a$gd h^hgd  & Fgd gd $a$gd L V ^ f n v ~  $ l p Z $$Ifa$gdU\l gd FfP $$Ifa$gdU\Z \ ` ( , v````$$Ifa$gdU\l kd# $$Ifl 0(# t0#644 lapytU\ |}WX xskskskssbb^gd  & Fgd gd kd$$Ifl0(# t0#644 lapytU\  xyRSYZ 9=>Ng~IMQRcg:>OPcdjXE=H h CJUVaJjHhwzh EHUjE=H h CJUVaJjh U h 6h=vh ; hflh h H* h H*hflh 6h = !"HIJKMh^hgd  & Fgd gd ^gd ?@ABCDEFhh^hgd gd  & Fgd gd /012mnopqrstugd  & Fgd h^hgd def} :>CDJKN^hpru36morvw$%ͻͻ洪旴旴旴洌hbHh CJaJh[h H*h[h 5H*\hech 5\ hech hbHh 5h)|h 5; h 5; h H*hflh 6 h 6h jh Ujhwzh EHU6,-./0123]^^gd 8^8gd gd  & Fgd h^hgd $%+,89WXYqr$IfgdU\l $$Ifa$gdU\l  & F gd gd ^gd ^gd $^a$gd +,789XY-HIyz汤朒xkgU#j;H hU\h CJUVaJhU\jhU\h EHU#j;H hU\h CJUVaJhU\hU\5hU\h 5H*hU\h 5jhU\h EHU#j;H hU\h CJUVaJjh U ha!]h hbHh hbHh 5 h 5h hbHh 6CJaJh 6CJaJ!+,-:DK`ygnkd$$Ifl !" t0644 lap ytU\$IfgdU\l $$Ifa$gdU\l yzH5$$Ifa$gdU\l $IfgdU\l kd+$$Ifl\( !TT  t0644 lap(ytU\0kd $$Ifl\( !TT  t0644 lap(ytU\$$Ifa$gdU\l $$Ifa$gdU\l $IfgdU\l 56JKcdwxyz汤撅sfj+hU\h EHU#j;H hU\h CJUVaJj(hU\h EHU#j;H hU\h CJUVaJj%hU\h EHU#j;H hU\h CJUVaJj hU\h EHU#j.;H hU\h CJUVaJhU\h jh UjhU\h EHU(F3$$Ifa$gdU\l $IfgdU\l kd$$Ifl\( !TT  t0644 lap(ytU\0kd"$$Ifl\( !TT  t0644 lap(ytU\$$Ifa$gdU\l $$Ifa$gdU\l $IfgdU\l 01F3$$Ifa$gdU\l $IfgdU\l kd#$$Ifl\( !TT  t0644 lap(ytU\14560kdX$$$Ifl\( !TT  t0644 lap(ytU\$$Ifa$gdU\l 6EFIJ$$Ifa$gdU\l $IfgdU\l JKc{F3$$Ifa$gdU\l $IfgdU\l kd%$$Ifl\( !TT  t0644 lap(ytU\{~0kd ($$Ifl\( !TT  t0644 lap(ytU\$$Ifa$gdU\l $$Ifa$gdU\l $IfgdU\l F3$$Ifa$gdU\l $IfgdU\l kd*$$Ifl\( !TT  t0644 lap(ytU\0kd-$$Ifl\( !TT  t0644 lap(ytU\$$Ifa$gdU\l        / 0 1 2 8 9 F G Z [ \ ] c d l m tb#j;H hU\h CJUVaJj6hU\h EHU#j;H hU\h CJUVaJj%4hU\h EHU#j;H hU\h CJUVaJj`1hU\h EHU#j;H hU\h CJUVaJj.hU\h EHU#j;H hU\h CJUVaJjh UhU\h &   $$Ifa$gdU\l $IfgdU\l   3 F3$$Ifa$gdU\l $IfgdU\l kd0$$Ifl\( !TT  t0644 lap(ytU\3 7 8 9 0kde3$$Ifl\( !TT  t0644 lap(ytU\$$Ifa$gdU\l 9 F ^ b c $$Ifa$gdU\l $IfgdU\l c d l F3$$Ifa$gdU\l $IfgdU\l kd+6$$Ifl\( !TT  t0644 lap(ytU\ 0kd8$$Ifl\( !TT  t0644 lap(ytU\$$Ifa$gdU\l  $$Ifa$gdU\l $IfgdU\l  6!<"G"["\"o"p"q"r"|"}"~"##]$^$q$r$s$t$$$%xxxxxxxxxx洧梛zusoggggjhU\Uht U h H*j>h h EHUj;H h CJUVaJ h H* h 6H* h 6jw<h h EHUjV;H h CJUVaJh4cJh 5;h 6CJaJhbHh 6CJaJhU\h jh Uj9hU\h EHU( 6!7!FAA<3^gd gd gd kd;$$Ifl\( !TT  t0644 lap(ytU\7!0"1"2"3"4"5"6"7"8"9":";"<"G"H"""5$6$7$8$%%%%^gd  & Fgd gd h^hgd  & F gd %%%%xxxxxxxxxxyyyyyyy$a$gd9C & Fgd ^gd Find another application for exponential functions or logarithms that was not mentioned in this lesson. What is the base of the function? Be prepared to share a brief explanation of how they are used to the class.     Log Function Exploration  FILENAME \p C:\Documents and Settings\John Kamkutis\Desktop\Topics Foundational to Calculus\Inquiry-Based Lesson\Team 3_Logarithmic-ExponentialInverseRelationship_CsarnyKamkutisMcKeen.doc Page  PAGE 6 of  NUMPAGES 7 xxxxxyyyyyyyyyyyyyyyyyyyyyyht hwCJ mHnHuh*CJ mHnHu h*CJ jh*CJ Uh*h2h9ChU\21h:pc/ =!"P#$% $$If!v h#v#v`#v :V l055`5 4ytU\Tkd$$IfTl d4 ttD`0((((4 laytU\T$$If!v h#v#v`#v :V l0, 55`5 4ytU\Tkd$$IfTl d4 ttD`0((((4 laytU\TDd @ 0  # Ab^(#mOi쓕tEvann^(#mOi쓕tEPNG  IHDRu[gAMA|QIDATH0 D PA.5_1jNݠ!n.a`@3 LdjyЇ2UzZ߆ՋCi.S*TxzߊiӋl& @TRuKf|tZql`>A{ 3 %#;{Ǟ{V,>!-cg4> +~ꣽ}ħM@<H@i6"IENDB`KDd d@b  c $A? ?3"`?2d=R,ӑHx ALqa`!id=R,ӑHx AL<^ 7xcdd``cd``baV d,FYzP1n:&B@?b u 8 ㆪaM,,He` @201d++&1l?z+8U`T T 4+ ;Hf%'"'?]bXE>F> Ir#K* 1f$;4BU~ca- CJρ^SøTA>| m&*EIENDB`$$If!vh#v#v:V l t0#655pytU\zDd @ 0  # Ab‘Z/F" an‘Z/F"PNG  IHDRu[gAMA|QpIDATHǽ˪ BUB^PU 9^jmg\\MEUTSUzwaT( Z7o c6Q> "ԗz 1wN ^jY3v\#!Bz$.(93A/#qjK>#1!ƭBdw/e¾4ȑ@׶y$m?@̈́@|5X}.j&1(Ɗ|Vlȃu>:a9,@^(RtqaKS2K%](IENDB`sDd @ 0  # Ab-çJYD'Tan-çJYD'TPNG  !"#$%&'()*+,-./0123456789:;<=>@ABCDEFmILMNQOPRSTVUWYXZ[\^]_a`bdcegfhjiknopqrstuvwxyz{|}~Root EntryZ F`OWK@Data ?xAWordDocumentY4|ObjectPool\A`OW_1211888426FAAOle CompObjfObjInfo !$'*-05:?DINSVY\^_`abcdfghijl FMicrosoft Equation 3.0 DS Equation Equation.39q5X] f(x)=log(x) FMicrosoft Equation 3.0 DS EqEquation Native Q_1211975156 FBBOle CompObj fuation Equation.39qt y= x  FMicrosoft Equation 3.0 DS Equation Equation.39qt y= x ObjInfo Equation Native  :_1211975000T FBBOle  CompObj fObjInfoEquation Native :_1211885976'FBBOle CompObjfObjInfoEquation Native  FMicrosoft Equation 3.0 DS Equation Equation.39qc_1211875528,;3FBBOle 3CompObj244fObjInfo56Equation Native 7>_12118755528FBBOle 8 FMicrosoft Equation 3.0 DS Equation Equation.39qX 10 0 FMicrosoft Equation 3.0 DS Equation Equation.39qCompObj799fObjInfo:;Equation Native <:_12118755766E=FBBOle =CompObj<>>fObjInfo?@Equation Native A:( 10 1 FMicrosoft Equation 3.0 DS Equation Equation.39qp 10 2_1211875587BFBBOle BCompObjACCfObjInfoDEEquation Native F:_1211875600@JGFBBOle GCompObjFHHf FMicrosoft Equation 3.0 DS Equation Equation.39qtb 10 3 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfoIJEquation Native K:_1211875612LFBBOle LCompObjKMMfObjInfoNOEquation Native P:_1211886678QF@B@B 10 4 FMicrosoft Equation 3.0 DS Equation Equation.39qFk] pH="logH + []Ole QCompObjPRRfObjInfoSTEquation Native Ub_1211886771OVF@B@BOle WCompObjUWXfObjInfoXZ FMicrosoft Equation 3.0 DS Equation Equation.39qK] M=logAA 0 ()Oh+'0 ,< P\ |   Equation Native [g1TableASummaryInformation([]DocumentSummaryInformation8el  IHDRu[gAMA|QiIDATHǵj0qk!UW**s |cf |m{7;rv5-l8Ah[eΚ0~Џk]F sT@J" 졎0ngQpbrKk8$Q vUT|j$/ٻI완6<\Im7@hXXM $I4+JI($&QdIrM€'DGnkȁ߹B' WPnBV+33IQ t%1 ^Kxa?ӱ@2F`y!8a|qð0.|b %FIENDB`$$If!vh#v#v:V l t0#655pytU\^Dd |b  c $A? ?3"`?2V\=L9Fקa`!|V\=L9Fק*`0JxmQ=KA}3{~XvX)I#De;KJaiORЀ? PS܏;1.þ}fgPf׀爉2DA4Ŋ?SVh~pH5]83N V=G!a`"c85 zq)a=Z:=Zsac9ұGrt0`? VFT;C7NS ,eo`DzУЙm;о2k ^WQ}aη_u慫ͦɓQ< 1UeFs1SC֗gew[K@oY\Dd |b  c $A? ?3"`?2js쟼ra`!zjs쟼r*`0HxmQJAf88-S|I#De;;)A,4 Rܟ;1.~37;2 &p0J\ &qe-Tώ(i2< ])tIů*6f("`#*6C/EYqnk47R r"u&]  c'Ysӽ8j[QZq$UO MnJBgve@R#V۫xud]N|0:]USuɨZwk݉bř!77&Y<Dd lb  c $A? ?3"`?2 b=]DpQFa`! b=]DpQX @Xhxcdd``^$d@9`,&FF(`TɁ A?d ?zjx|K2B* R0pE1At 2BaR`b Y7@rȁL@(\PX$U@pT9LF]F\ 8AZWB!P*F[%L69\=700*v1.Oefgv\! ~ Ay b.-h74$̫pBÅ rSa! Qn3#E>c2fȞ'.hS v 0y{)I)$5bPd+I~,İrt_3s$$If!vh#vx!:V l t065"p ytU\$$If!vh#v0#v2#v:V l t0655T5 p(ytU\&Dd @b   c $A ? ?3"`?2pF8^AuAL+a`!DF8^AuA@  xcdd`` @c112BYL%bpugMa(w rr'\F{WW׀ *\;& `pX321)Wx\ ]` bg!"Z$$If!vh#v0#v2#v:V l t0655T5 p(ytU\&Dd @b   c $A ? ?3"`? 2p]ahiwLa`!D]ahiw@  xcdd`` @c112BYL%bpugMa(w Dprr'\F{WW׀ *\;& `pX321)Wx\ ]` bg!Y [^$$If!vh#v0#v2#v:V l t0655T5 p(ytU\%Dd @b   c $A ? ?3"`? 2ow3B7wK a`!Cw3B7w@  xcdd`` @c112BYL%bpugMa(w Z˝s] _].hpC\h0y{aĤ\Y\ q1(2t)v<^f~W\D$$If!vh#v0#v2#v:V l t0655T5 p(ytU\$$If!vh#v0#v2#v:V l t0655T5 p(ytU\$$If!vh#v0#v2#v:V l t0655T5 p(ytU\$$If!vh#v0#v2#v:V l t0655T5 p(ytU\4Dd 0@b   c $A ? ?3"`? 2~U<{1';vZ&a`!RU<{1';v k  xcdd``fed``baV d,FYzP1n:&f! KA?H1 ہqC0&dT20ͤ KXB2sSRs=^~*cF\_2VS|=8_7< @AahaMa(w Hs\FV_׀ j\;f `p\021)Wx\ ]` g!G`S$$If!vh#v0#v2#v:V l t0655T5 p(ytU\'Dd @b   c $A ? ?3"`? 2qVCg n`M)a`!EVCg n`xt xcdd`` @c112BYL%bpu 4CF[*,.a`!$> 4CF[*`0 xcdd``> @c112BYL%bpu 1 @c112BYL%bpu u;&LLJ% "CD1|,İr!v120eqJ?$$If!vh#v0#v2#v:V l t0655T5 p(ytU\Dd |@b  c $A? ?3"`?2PtHqY^t3+,i4a`!$tHqY^t3+`0 xcdd``> @c112BYL%bpu 1 @c112BYL%bpu 1 @c112BYL%bpuVf +'?= ~i l040y{I)$5bPd(/v<^+f,0dfDd xb  c $A? ?3"`?2|yeNz $©P-?a`!|yeNz $©P `\{xڥKP]_`""M)*uBl.V*4Fh!d(7Dp#APX0޽lqr]8HWV J`ER-i}6e14tBrf `NiIRW NN// ErO6IVnS\g_* Ds>kί~Ho|)lm[?'8eSqOP k ?eSSW9DWՏL꧶:M{~P>_6)Yg!?!y'D[1 R o,U񳱒 ;\,=;/Zrt)'R Wfq, 5W]KCz(TActivity: Analyzing the Relationship between Logarithmic and Exponential Functions.Normal reese gray5Microsoft Office Word@ @М̨@&3@K{՜.+,0< hp|  4 TActivity: Analyzing the Relationship between Logarithmic and Exponential Functions Title  F Microsoft Word 97-2003 Document MSWordDocWord.Document.89qs2&6FVfv2(&6FVfv&6FVfv&6FVfv&6FVfv&6FVfv&6FVfv8XV~ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@ 0@_HmH nH sH tH @`@ [NormalCJ_HaJmH sH tH B@B  Heading 1$h@&^h5\:@:  Heading 2$@&5\DA D Default Paragraph FontVi@V  Table Normal :V 44 la (k (No List :B@: Body Text$a$5\4@4 Header  !4 @4 Footer  !j@#j a!] Table Grid7:V0B^@2B ec Normal (Web)dd[$\$4UA4 ec Hyperlink >*phHRH | Balloon TextCJOJQJ^JaJPK![Content_Types].xmlN0EH-J@%ǎǢ|ș$زULTB l,3;rØJB+$G]7O٭VvnB`2ǃ,!"E3p#9GQd; H xuv 0F[,F᚜K sO'3w #vfSVbsؠyX p5veuw 1z@ l,i!b I jZ2|9L$Z15xl.(zm${d:\@'23œln$^-@^i?D&|#td!6lġB"&63yy@t!HjpU*yeXry3~{s:FXI O5Y[Y!}S˪.7bd|n]671. tn/w/+[t6}PsںsL. J;̊iN $AI)t2 Lmx:(}\-i*xQCJuWl'QyI@ھ m2DBAR4 w¢naQ`ԲɁ W=0#xBdT/.3-F>bYL%׭˓KK 6HhfPQ=h)GBms]_Ԡ'CZѨys v@c])h7Jهic?FS.NP$ e&\Ӏ+I "'%QÕ@c![paAV.9Hd<ӮHVX*%A{Yr Aբ pxSL9":3U5U NC(p%u@;[d`4)]t#9M4W=P5*f̰lk<_X-C wT%Ժ}B% Y,] A̠&oʰŨ; \lc`|,bUvPK! ѐ'theme/theme/_rels/themeManager.xml.relsM 0wooӺ&݈Э5 6?$Q ,.aic21h:qm@RN;d`o7gK(M&$R(.1r'JЊT8V"AȻHu}|$b{P8g/]QAsم(#L[PK-![Content_Types].xmlPK-!֧6 0_rels/.relsPK-!kytheme/theme/themeManager.xmlPK-!R%theme/theme/theme1.xmlPK-! ѐ' theme/theme/_rels/themeManager.xml.relsPK] | ((   d xy$08=L Z y16J{ 3 9 c 7!%y !"#%&'()*+,-./12345679:;   O c e cwy/1FZ\l[oq]qs:::::::::::::::::(6!8@0(  B S  ?<$<$<$<$<$<$<$<$D D tH H y9*urn:schemas-microsoft-com:office:smarttagsplace8*urn:schemas-microsoft-com:office:smarttagsCity -.9ALO6: yz56JK 89cd7;-."(= ? f yz56JK 89cd3333333 y^ Fi:xd$Ry t.2eA '[5=vz.* 9~VU0= hkc)|9@@@ @@UnknownG.Cx Times New Roman5Symbol3. .Cx Arial5. .[`)TahomaA$BCambria Math"1hM9gP97K{4K{44 2QHP?U02!xx SActivity: Analyzing the Relationship between Logarithmic and Exponential Functions. reese gray4         CompObjkr