Lesson 5 Base Change Formula

Lesson 5 Base Change Formula - Exponential& Logarithmic...

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Unformatted text preview: Exponential & Logarithmic Functions Lesson #5: Changing the Base of Logarithms Warm- Up #1 In the previous lesson we discussed logarithms with many different bases. In order to do numerical calculations on a calculator, we use two specific bases - base 10 and base e. Common Logaritth Common logarithms are logarithms in base 10, eg. logwlOOO. These logarithms are in such common use that when a base is not given the logarithm is understood to be in base ten. For instance, log101000 is often written as log 1000 On a graphing calculator, this can be evaluated using the key. Class ax. #1 Evaluate each of the following logarithms manually and by calculator. 3) 10391000 b) logwxs/ 1000 "so; Manqu-J \V‘) mama-gag . we” 3 1-900%. 3 a ((0) (a? ([000) " 1,3(000 69” ’° 1 1.131000 £3" 1"“ ((0‘)’ Ant». 2. .2, by" l 'L 41,. L103 Natural Logaritth A” I Natural logarithms are logarithms in base 2 eg. logeIS. e is an irrational number, the value of which will be determined in Warm-Up #2. logels is often written as In 15 On a graphing calculator, this can be evaluated using the key. ' Class Ex. #2 l Evaluate the following logarithms to one decimal place where necessary. ‘-”'A 1i a) ln5 b) logeS c) lne It UM ‘ a..— 122 Exponential and Logarithmic Functions Lesson #5: Changing the Base of Logarithms Warm-Up #2 is not required for this course, but may give students a greater understanding in preparation for higher level math courses. Warm- Up #2 Approximating the Value of e The formal definition of the irrational number e is the limit as x approaches infinity of the 1 x function fix) = (I + E) . Complete the following table to determine the value of this function as x gets very large. Use the TABLE feature of a graphing calculator and work to 4 decimal laces. 7‘ P Y. = t l + 3%) wow moo x Estimate for e is ‘1' 7’93 A more accurate estimate can be determined by pressing the E key on a graphing calculator. The value ofe to 9 decimal places is 3" 3“! 961 8! 938 Warm- Up #3 a) Evaluate log525 2. '32 : 1075 61 5 ' a» 2 5‘5: 52' 3a.?) b) Try to evaluate log550. What problem, do you encounter? 5“: 5e Base 5 “En cm Expenen‘t (whole number) does m‘t {auntie ‘Eu 56 . At the moment we are unable to evaluate log550, but by converting to common logarithms, or natural logarithms, we can use a calculator to determine the value of log550. The method for converting from one base to another is discussed in Warm-Up #4 Exponential and Logarithmic Functions Lesson #5: Changing the Base of Logaritth 123 Warm- Up #4 Change of Base a) Evaluate. , ,, log25 _ 10ge25 _ 3.2198 1)10g525§3z 52 ll) logs '___’_2____ 1“) logeS - I.‘ 09 7= 1 13—. b) Evaluate. , ,, 10g 243 . 5 10ge243 _ 5 1) log32433,= 3, 11) log 3 -_= 111) loge?) - = 3: 5 c) Write log264 in a form which can be evaluated using a calculator. _ My Input Haj: ,cojtw) + 103(1) at” Change of Base Identity logac logbc = logab This formula is NOT on the formula sheet We have seen in Warm-Up #4 that the above identity is true for converting logarithms to base 10 or base e. In fact it holds true for converting logarithms to any base. The example below supports this. 1 1024 4 i) Evaluate log41024 ii) Evaluate 0:2—4 : £2" ("34.) 2:211 - M g2 17(4) ? :2 ' W = 4;— :_—_§~_ : 5 We will be able to prove this identity in a later lesson when we have developed an understanding of the laws of logarithms. Class Ex. #3 Evaluate the following logarithms to the nearest hundredth by changing the base. @ a) log5221 b) logzfi c) 310g7512 2-! _ (Hum) _ L: {I '43:; 3%) 5 = —q.97 : 38.1058...) : iéi 124 Exponential and Logarithmic Functions Lesson #5: Changing the Base of Logaritth Convert the following logan'thms to the base indicated. a) log6216 tobase3 b) log 300 tobaseS - {$314 - E2220 ' .jegsé ' L375 [0 l b) 2log8512 c) -log7(§a§) - E z Ki} 1 ’ flaw-.73) 1% 5 It; 7 =®€5>¢ = +32% 5: 1 —5 e) logz m f) log-,49 ‘5 _ aims-g.» = “Eli—("i ) ’ 10? 2 g In; 7 : — 5 ': - 1' 0 Complete Assignment Questions #1 - #10 ASSIgnment 1. Evaluate each of the following logan'thms. a) log 100 b) logblO6 c) log\/ 10 E d) log 0.01 :9...“ slugs: a2?“ Us); '5 J. 2. Evaluate the following logarithms to the nearest tenth. =13 a) In 20 b) logeS c) lne2 3.0 gfi = 3‘] .2. o m a suns; *- 3. Convert the following logarithms to the base indicated. a) log335 to base 7 b) log 5- to base 6 c) log350 to base e .. £732.. _ j” l f: _ m 139; 8 [9:6 ([0) Jazz (3) Exponential and Logarithmic Functions Lesson #5: Changing the Base of Logaritth 125 4. Evaluate using the change of base identity to the nearest hundredth: a) log517 b) logo‘s 5.9 c) 10; 3 d) ~210g126 e) log88 5 = M : 2.5gng 3 % v2 A-é _ axis is; 5 {aim £3 1‘7” £67 3 “f = Wk 1 32, 5g ‘- "W" a law we -_.__ 3 flag: m 5. Evaluate each expression: j a) 410344 b) 1010g101000 : 1-}, : [03 2‘ ‘7’ :2 I030- Multiple 7. Which of the following has a negative value? ‘ Choice A. —log4(0.1) B. log4(—:-) C. log% D, log4(-§-) A“ case (:58 ~64?- Numerical 8. The value of the expression log fl 8 + 210 93 to the nearest tenth is 7' a . Response to 8 ___L. + j s o h u - O 9. Given the equation log7x =10? , t e value of x to the nearest whole number is 3’ 3 . ,1: 4152. 1964 10131 = ?5’3... Joflsws'... z = 7 § : g 10. If logx27 = 10g123, the v ue 0 x o the nearest whole number is ’723 . @2931 i {a}. 1%? 32315,, . ‘2, 5“- aamz» affixed is 5‘ lay/cl: ' ? ll) 1: [7:18 126 Exponential and Logarithmic Functions Lesson #5: Changing the Base of Logarithms IHISVVBr'lflEY 1 1. a) 2 b) 6 c) E d) —2 2. a) 3.0 b) 2.1 c) 2.0 1 3 ) 10g735 ) 10g6(§) ) logeSO . a c log78 10g610 loge3 4. a) 1J6 b) —256 c) L46 d) —L44 e) L00 5. a) 4 b) 1000 logx . . 6. Graphy=—, x-1ntercept1s1 7. D 8. 7.0 9. 313 10. 1728 log3 ...
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Lesson 5 Base Change Formula - Exponential& Logarithmic...

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