Math 111! Name:__Key__________________________________________________________________Grp!#________!GrpAct – #17!! ! ! ! Logarithmic!Functions!! ! ! !!!!!!!!! !!!!!!!!!Sects!5.4!&!5.5!!Prerequisite!Skills!Key!Terms!Learning!Outcomes!• working!with!inverse!function s!• composing!functions!with!their!inverse!• working!with!exponential!functions!• !• Logarithmic!Function! !• Natural!Logarithm!• Common!Logarithm!! !• Logarithm!Properties!• Inverse!Prope rtie s !fo r!L og s!• Change!of!Base!Property!• Exact/Approximate!solutions!• Convert!between!exponential!and!logarithmic!statements!• Evaluate!logarithmic!expressions!• Apply!change!of!base!formula!to!logarithmic!expressio n s!• Use!logarithms!to!solve!exponential!equations!• Use!exponentiation!to!solve!logarithmic!equations!• Determine!domain/range!of!logarithmic!functions!• Apply!the!properties!of!logarithms!to!simplify!or!expand!logarithmic!ex p res sio n s!• Model!with!logarithmic!functions!1. Last!class!we!talked!about!exponential!functions.!Let!f (x) = 4x!and!g(x) =15⎛⎝⎜⎞⎠⎟x.!!Evaluate!each!of!the!following!a. !!g(2)=125!! ! ! b.!!!!f (0.5)= 2! ! ! c.!!!!f (−3)=164!d.!!!!!g(−2)= 25! ! ! e.!!!!f (0) = 1! ! ! f.!!!!!g(0) = 1!!2. This!class!we’ll!look!at!logarithmic!functions.!!In!your!MML!HW!a n d !R e a d in g !you!saw!!how!exponential!and!logarithmic!equations!are!related.!!Fill!in!the!table!with!the !!alternate!forms!of!each!equation.!!!Exponential!Form!Logarithmic!Form!Written!Form!5−2=125!51log 225⎛⎞=−⎜⎟⎝⎠!The!base!5,!to!the!–2!power,!is!equal!to!125!328=!log2(8) = 3!The!base!2,!to!the!3!power,!is!equal!to!8!!43= 64!!log4(64) = 3!The!base!4,!to!the!3!power,!is!equal!to!64! 811/4= 3!log81(3) =14!The!base!81,!to!the!!14!power,!is!equal!to!3!!7−1=17!log717⎛⎝⎜⎞⎠⎟= −1!The!base!7,!to!the!–1!power,!is!equal!to!!17!!1001 2= 10!1001log (10)2=!The!base!100,!to!the!!12!power,!is!equal!to!10!!130= 1!log13(1) = 0!The!base!13,!to!the!0!power,!is!equal!to!1!!by= x!logb(x) = y!The!base!b,!to!the!y!power,!is!equal!to!x!!!e3.4657= 32!ln(32) 3.4657≈!*see!note!below!The!base!e,!to!the!3.46 5 7!power,!is!equal!to!32!!p!is!the!Power!that!!goes!on!!the!!Base!b!!to!give!you!m.!!!!!! ! ! !!!You!may!have!already!realized!this,!but…!If! ,!then!!The!inverse!function! o f !a n!exponential!function!is!a!logarithmic!function.!!*Don’t!know!what!ln !means?!It!is!the!logarithm!w ith !b a se !e,!ln(x)!=!loge(x).!This!logarithm!is!u sed !a!lo t!in!m ath ematics,!and!it!is!called!the!natural!logarithm.!Do!you!remember!last!week!when!we!saw!exponential!functions!with!base!e?!Those!were!called!natural.exponential.functions!!3. The!domain!and!range!of!logarithmic!and!exponential!functions.!a. What!are!the!domain!and!range!of!f(x)!=!bx?!!!D = (−∞,∞) R = (0,∞)!!!!b. Explain!why!the!equation!ex!=!0!does!not!have!any!solutions.!!A!positive!based!raised!to!any!power!is!greater!than!zero.!!!!ex= 0 ⇔ e = 01 x= 0BUTe ≠ 0 therefore ex≠ 0 for any value of x!!!!c. Try!to!evaluate!ln(0).!Explain.!Remember:!if!pme=,!then!ln( ).pm=!!!!ln(0)= z ⇔ ez= 0!!!So!by!part!(b),!!!ln(0)!Does!Not!Exist!(DNE)!! !!!d. Explain!why!the!equa tion!515x=−!does!not!have!any!solutions.!!A!positive!based!raised!to!any!power!is!greater!than!zero.!!!e. Try!to!evaluate!log5(-15).!Explain.!Remember:!if5pm =,!then!5log ( ).pm=!!!!log5(−15)= z ⇔ 5z= −15!!!So!by!part!(d),!!!log5(−15)!Does!Not!Exist!(DNE)!!!!f. What!is!the!domain!and!range!of!log ( )bx?!!The!function!log ( )bxhas!the!same!do m ain !and !ran ge,!regardless!of!the !base!b.!!!!D = (0,∞) R = (−∞,∞)!! !Natural!and!Common!Logarithms!loge(x)!is!called!the!natural.logarithm!and!is!written!ln(x).!!!log10(x)!is!called!the!common.logarithm!and!is!written!log(x).!!4. On!the!axes!below!sketch!a!graph!of!y!=!ln(x)!and!y!=!log5(x).!Indicate!which !gra ph !is!wh ich .!Start!by!plotting!a!couple!of!points!on!each!graph.!Label!the!x-intercept(s)!on!each!graph.!!(Do !NO T !us e!yo ur!ca lcula tor!to!g et!the !grap h .)!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!5. Evaluate!each!logarithm!and!write!down!a!written!statement!of!what!it!means.!!!! Recall:!y = logbx ⇒!!“y!is!th e !p ower!that!go es !o n !th e!b ase!b!to!give!you!x.”!! ! ! ! !a. log3(81) = 4! ! ! ! c.!!!!! ln e9( )= 9!!b. log1100⎛⎝⎜⎞⎠⎟= −2! ! ! d.!!!! log33−7( )= −7!x!y!=!log5(x)!125=152= 5−2!!log5125⎛⎝⎜⎞⎠⎟= log5(5−2)= −2!!15= 5−1!!log515⎛⎝⎜⎞⎠⎟= log5(5−1)= −1!!1 = 50!!!log51( )= log5(50)= 0!5 = 51!!log5(51)= 1!52!!log552( )= 2!53!!log553( )= 3!x!y!=!ln(x)!1e2= e−2!!!ln1e2⎛⎝⎜⎞⎠⎟= ln(e−2)= −2!!1e= e−1!!!ln1e⎛⎝⎜⎞⎠⎟= ln(e−1)= −1!1 = e0!!!ln 1( )= ln(e0)= 0!e = e1!!!ln e( )= ln(e1)= 1!e2!!!ln(e2)= 2!e3!!!ln(e3)= 3!Inverse!properties!o f!logarithms!!!!6. Using!Logarithms!to!solve!Exponential!Equations.!The!box!at!the!right!gives!two!very!important!properties!of!logarithms.!If!you!loo k !b a ck !at!the!previous!problem,!you’ll!see!you!were!actually!using!the!first!inverse!property!on!pa rts!(c)!and !(d)!!!That!first!inverse!rule!gives!us!a!way!to!solve!expo ne ntial!eq ua tion s.!! Use!it!to!so lve !t he !e q u at io ns !b elo w;!give!the!Exac t!solution!for!each!and!the!approximate!solutions!if!possible.!!a. 10 210x=! !!!! x = log(210)! 2.3222!!!!!b. 215xe=.!!!! x = ln(105)! 4.6540!!!!!c. 25 13x=!!!!! x =12ln(13)ln(5)⎛⎝⎜⎞⎠⎟! 0.7968!!!!!!7. Using!exponentials!to!solve!logarithmic!equations.!To!solve!a!logarithmic!equation!we!use!the!property!in!the!box!below!and!the!other!inverse!property!stated!in!the!box!near!the!top!of!the!page.!Check!your!solutions.!!a. log(5 ) 2x =!!!!!x = 20!!!!!!!b. 3ln( ) 12x =!!!!! x = e4! 54.5982!!!!!!Exact!and!Approximate!values.!is!an!exact!value.!!is!an!approximate!value.!is!an!exact!value.!is!an!approximate!value.!!Note!tha t! some!logarithms!can!be!simplified!exactly.!For!example,!!log3(9)!=!2!is!an!exact!value.!Using!Exponentiation!to!Solve!an!Equation!If! !then! !for! !The!above!statement!is!true!because!when!we!input!equivalent!quantities!into!a!function,!the!resulting!outputs!are!also!equal.!By!carefully!selecting!the!value!of!b!we!can!use!this!idea!to!solve!a!logarithmic!equation!like!those!to!the!left.!!After!“exponentiating”!both!sides,!we!can!use!the!inverse!rule! . !Important! P rope rt ies!of!Logarithms!!Produc t!La w !!Quotient!Law!!Powe r!Law !!!!For!derivations!of!these,!se