Name Date. Properties of Logarithms For use with Exploration 3.5. The corresponding exponential forms of these two equations are. = n.

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1 Nae Date.5 Properties of Logariths For use with Eploratio.5 Essetial Questio How ca ou use properties of epoets to derive properties of logariths? Let = log ad = log. The correspodig epoetial fors of these two equatios are = ad =. 1 EXPLORATION: Product Propert of Logariths Work with a parter. To derive the Product Propert, ultipl ad to otai = = +. The correspodig logarithic for of = + is log = +. So, log =. Product Propert of Logariths EXPLORATION: Quotiet Propert of Logariths Work with a parter. To derive the Quotiet Propert, divide to otai = =. The correspodig logarithic for of = is log =. So, log =. Quotiet Propert of Logariths EXPLORATION: Power Propert of Logariths Work with a parter. To derive the Power Propert, sustitute for i the epressio log, as follows. log = log = log = ( ) Sustitute for. Power of a Power Propert of Epoets Iverse Propert of Logariths 117 Copright Big Ideas Learig, LLC

2 Nae Date.5 Properties of Logariths (cotiued) EXPLORATION: Power Propert of Logariths (cotiued) So, sustitutig log for, ou have log =. Power Propert of Logariths Couicate Your Aswer. How ca ou use properties of epoets to derive properties of logariths? 5. Use the properties of logariths that ou derived i Eploratios 1 to evaluate each logarithic epressio. a. log 1. log 81 c. l e + l e d. 5 le le 5 e. log5 75 log5 f. log + log Copright Big Ideas Learig, LLC 118

3 Nae Date.5 Notetakig with Vocaular For use after Lesso.5 I our ow words, write the eaig of each vocaular ter. ase properties of epoets Core Cocepts Properties of Logariths Let,, ad e positive real uers with 1. Product Propert log = log + log Quotiet Propert log = log log Power Propert log = log Notes: Chage-of-Base Forula If a,, ad c are positive real uers with 1ad c 1, the log a log c a =. log c I particular, log c a loga = ad logc l a log c a =. l c Notes: 119 Copright Big Ideas Learig, LLC

4 Nae Date.5 Notetakig with Vocaular (cotiued) Practice A Etra Practice I Eercises 1, use log 5. ad log to evaluate the logarith. 1. log 0. log 1 1. log 1 5. log 70 I Eercises 5 8, epad the logarithic epressio. 5. log 10. l 7. log 8. l I Eercises 9 1, codese the logarithic epressio. 9. log + log log5 log l + l log 5 log log 7 1. log log + log 1. Copright Big Ideas Learig, LLC 10

5 Nae Date.5 Notetakig with Vocaular (cotiued) I Eercises 1 17, use the chage-of-ase forula to evaluate the logarith. 1. log log log log For a soud with itesit I (i watts per square eter), the loudess L(I) of the soud (i deciels) is give the fuctio LI ( ) = 10 log I, where I0 is the itesit of a arel audile soud (aout I watts per square eter). The itesit of the soud of a certai childre s televisio show is half the itesit of the adult show that is o efore it. B how a deciels does the loudess decrease? 19. Hick s Law states that give equall proale choices, such as choices o a eu, the average hua's reactio tie T (i secods) required to choose fro those choices is approiatel T = a + log ( + 1) where a ad are costats. If a = ad = 1, how uch loger would it take a custoer to choose what to eat fro a eu of 0 ites tha fro a eu of 10 ites? 11 Copright Big Ideas Learig, LLC

6 Nae Date Practice 5. BPractice B I Eercises 1, use log5 0.8 ad log to evaluate the logarith. 1. log5 81. log 1 5. log 1 5 I Eercises, epad the logarithic epressio.. 7 log 1 5. log 5. log8 7. Descrie ad correct the error i epadig the logarithic epressio. I Eercises 8 11, codese the logarithic epressio log9 log9 9. log log8 10. l + 5 l + l 11. log 9 + log 1 log I Eercises 1 1, use the chage-of-ase forula to evaluate the logarith. 1. log log 0 1. log Your fried clais ou ca use the chage-of-ase forula to write the epressio l as a logarith with ase. Is our fried correct? Eplai our reasoig. l 1. For a soud with itesit I (i watts per square eter), the loudess LI ( ) of the soud (i deciels) is give the fuctio ( ) 10 log I LI, I is the = where 0 I0 1 aout 10 watts per square eter. The ass itesit of a arel audile soud ( ) guitar plaer i a ad turs up the volue of the speaker so that the itesit of the soud triples. B how a deciels does the loudess icrease? Copright Big Ideas Learig, LLC 1