Week 4 - Lifts IThe e(5.2 2 E:2 I Note The base e is called...

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Unformatted text preview: Lifts IThe ExponentialFunctionylr; e (5.2] 2 E :2 I Note: The base e is called the natural base and for our purpos e is an irrational number which isf-Japproximatelygequal to 2,7182818284 ...... .. ' The base e is sometimes easier to use than the other base Graph y‘= b" ,l b > 0 by remembering the general lool< of an exponential equation 5’9? 7 ‘ ' g x .. . ‘7 l , .1' a e I i I’ I . 1 \ .Range; ‘ 'y-intercept: (0' l) x-intercept: No we Asymptote: H‘Q‘fi 2’0 h‘l'l‘vc MN)th ' ‘j 1: O . ..._..Maw“......._:__..u.._;flem;.._._.,...lfiz.wmmmm;.._ How do we define e to be 2.718E8 ..... .. I. x ' Let’s investigate the values of function f(x) =[1+(—)] .as X gets larger and larger fl. - - x m Thslnniewm Observe: As it gets bigger and bigger the value of fix) approaches flvwd W % W96, Theorem : Let a be any rEaI number. The instantaneous rate of cha e._of the s; function f(x) = e” at x = a is Q So the instantaneous rate of change of the function K f(x) =.ex at the point (a ,e“) is ea. This property is unique to the exponential cg '3 @ WZQ function (x) = ex or constant multiples of the exponential function such as , ea f(x) : kex where k is a constant. ’ You will have to remember and use this theorem fre uentl 1 when ou take Calculus. \ EX: In a biology experiment, let N t b - the number of bacteria in a colony after t hours, where z 0 corres onds to the . ime the ex eriment be ins. Suppose that during the period'from t ; 4 hours to hours the number of bacteria is modeled by the experimental function N [t] = 4 et : +. ' 46:0: +[Ii=‘i t=4 , Mm:- 4'8 t=<3 , «9:4? L'. t 55:37 x1 KZ‘ t: c] Use your knowledge about the exponential function to compute - =7 Instantaneous Rate of Change of the population at the instan :9 mg I W4) GAS—4 3592’ Nowifihvfi (baa kQO aflgarirmrfiflnction (3:; .5; I Yes Recall the general graph of the exponential function y = 2x. This function passes the horizontal line test. Thus it is one to one and has an inverse. Procedure for finding the inverse of f[x) or f ‘1(x) In this case y Definition: The logarithm base b_ of f. , which b must be raised to yield x. I I ' Another words, This relationship allows one to convert logarithmic equation to exponential equation and vice versa. m9 x so» 2 a“ fix): a “39%. \nex Ex: Now let’s sketch the graphs of y = logax, y 2 iogx and y = lnx on the same set of axes. You can figure out the logarithmic graphs from their corresponding exponential graphs by treatinw ; . . _ __ verses EX: Sketch where b is any positive constant by specifying the domain, range, asymptote and inter ept NO V-A- "0 Jim" ‘ . {1’ ~ ! fl+ Note. The domain of any 1 - function is (0,00) i.e. only positive x value ‘ CD,i\ - . ii fir ‘ \fiAt QM ;O N ‘ a EX: Find the domain of . w W'- . ii 5% Ex: Graph y =-10g3( +1 y remembering shifting and reflecting. Specify the drarige, n (are? asymptotes. g 35” OFHO“ 5» YD“ Domm n‘ g Mva {-0 Lab ‘ KKKK \ u E - cide which quantityis larger? X I a z ‘7 Mme - 519613 ~ ' 3.1%., '0 log32 n r _ 2/ 33m .‘ 3 Remember the conversion equation between log and expOnential equation. Evaluate: 1) logIO '2 In general log b [2 =1 Properties of Logarithms (5.4) Rules: 1) a. leggy? ' logbl: O 2) 10gbé’Q)‘ + ex: Simplify log50+logZ=' baffled) 1 “1:: log ([00) >2; 7, 10/" MW) 3] 10gb%: We?”l’0€lb(—§K ' ' H ex: Simplify logs 56— logs 7 T: 2%? 7:1, 4) 1°ng = V1 we). ex: Simplify logfi/IE b \ :5 w; (M; 1% (241% 4631(1)? 54/; has; :(g‘nm 2:;- 1 1‘ 5) 1 for all real X. - \9 6] bl°gix= forx>0 I _ I . :z; 'X ' lo g: w v) 13‘ X70 For the rest of the section we will learn how to apply the above properties. M _ _ (pH/\sz/ Ex: WriWression as a single 10 arithm with coefficient 1: 1 $ . logb4+ 3[10gb(1+x)——.iogb(l —x)} :1 ‘ - 2 Dog 4} +LU 0+1} ! 3b WM ) mar-23 E3 MM 6%} 1293 5x6) EX: If 10gb 2 z A, 10gb?) = B, 10gb5 =.C Write log 5,03 in terms of A, B, C. EX: Write 10gb 85/3 in terms 01A? ) ‘ \ [139552 ' l ' T. ‘8' 39: -: 00% babblz I b i 3 mafia ' z :5. 3b2 H/z L' :. 3A +V2, * . ’1 EX: Solve the following equations using the log properties. SQ . X‘ P. I ' 2)1;)'2:Q+3=280 ' \qke Law! t9? 114. :. [NJKH v {SE/53:3”: «— ~3 " *2, QKTW A Change of base formula: (3 0b MK :— 2 - °: 22’ K WWW? “WWW 87‘” Q --” t - q c302? "Rh/tel at!“ X Ex: Convert in decimal form 10g210 = : b 9 > F g 3:6” 6. legal? 2 12 implies ab = C LDMVMRWM “GmnuLA ? 7. log5 24 is between 51 and 52 T 5', *2: 2L} 4 X Z7. /- 8. 10g524 is betweenland2 E ' I W L M 9. The domain-0f lnx is the set of all real numbers X96”! Are the following properties True/ False? Use log properties to justify your answer. 1‘1- AB T1.10A+10B—0C=10(-—) Where as to solve the exponential equation m ‘ 5" = 32"“1 we will have to use the fact 41100.2 8 OJ}. wwwmg '2; (l: 1933 “'" 1 124’s? A I}: ” “\q’ MAMMW‘ L1 .2 UN; 4 -‘ ‘5’,\_— 5-15 Equations and Inequalities with Logs and Exponents (5.5 | 1; 0 2(1 H) :29: +2 - mudilimwt Em lawn/H r Two facts you should be always aware of while solving equations with logarithmiobe Wge l‘ W and exponential expressions: X ,. " ') - be, W “L 1)The difference between'a conditional equation‘an'd an identity. For example, 'W [l W; 2" l3 = 4’? is called a conditional equation because AW- ‘ SQM§QJEW ‘ "w" M1“ 1)Since exponetial equaticjfyfine to one [because it passes the horizontal line But the above facts are only true if the base b of both sides of the equation'is the same. But if the bases are different for the exponential equations, we have to take log or In function'of both sides and use the log propertiesto solve the given equation. For example, in order to solve . k 2"+3 = 4" we will have to use the fact that Another common example of exponential equation in quadratic form that needs a Clever approach is as follows: Solve: 3(23“)-11(23‘-)—4,= 0 %mmnm.mm.._m_.li.nmm.__ Now let’s spend some time solving equations in logarithmic form: Recall: X ; Yimplies bY = X [conversion definition from sec 5.3] We will be using this fact'very frequently to solve the following eqLiations: 50mm _ I I M [my $67ij Ex: log2(x+1)+ log2(x + 5) i 5 95114 H \\ (X-‘rfl .7 [Hm $03“; ) "3* 2g "XL—ksm‘t’fi‘if 7’. 3?— XfL—pr +§9 : ’3’)? ‘I (L H.7Nolef KW . F omen ‘60“ "(Kth (“'Bl "0 0% NW A I l l... ‘5 30 {Ducalml ms ( MW” Ex: ln(x+1)=2+ln(x 1)‘ ... *QM I“ «Q/m[x4l\*,Q/u[w(~x3: Z Example of an inequality: X 7 o 36H 70 935 W 70 i I f Solve for X: log2 x + log2(x + 1) — logzmx + 6) < 0 7...: W a. ‘3 . m.- I a Note: First look at the original domain before start solving the inequali X ? m3 0 mm m '. A i! “z, 7 Q _ K7 0 JZWL an 'I%I)+,Q,ngl(fLX—Hfl <0 7 C {1/05 * 33 g L 2,. - Y 3-“ 3 b _ fifkbgfigcflfi‘eefl 2; ...
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