MA Lesson 16 Notes Summer 2016 Properties of Logarithms. Remember: A logarithm is an exponent! It behaves like an exponent!

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1 MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more properties for logrithms. Since logrithms ehve like eponents, you should see similrity etween the properties of eponents nd the properties of logrithms. I PRODUCT RULE m n m n Product Rule for Eponents: (when powers re multiplied, eponents re dded) Product Rule for Logrithms: log M log M log (the logrithm of product is the sum of the logrithms) Informl proof of product rule: log00 = log000 = log(00000) log log00 log000 When single logrithm is written using the product rule, we refer to this process s epnding the logrithm or epnding the logrithmic epression. (Becuse we cn only find logrithms of positive vlues; when using these properties of logrithms, we ssume ll vriles represent positive vlues.

Since logrithms ehve like eponents, you should see similrity etween the properties of eponents nd the properties of logrithms.

2 MA 5800 Lesson 6 otes Summer 06 E : Use the product rule to epnd ech epression nd simplify where possile. ) log 7r ) log ( y) log(00 ) d 5 ) ln(0 e ) II QUOTIET RULE m mn Quotient Rule for Eponents: n (when powers re divided, eponents re sutrcted) Quotient Rule for Logrithms: log M log M log (the logrithm of quotient is the difference of the logrithms of the numertor nd the denomintor) CAUTIO: log ( M ) log M log log M log log M We cn lso epnd logrithm using the quotient rule. Agin, ll vrile re ssumed to represent positive vlues.

sutrcted) Quotient Rule for Logrithms: log M log M log (the logrithm of quotient is the difference of the logrithms of the numertor

3 MA 5800 Lesson 6 otes Summer 06 E : Use the quotient rule to epnd ech logrithm nd simplify where possile. 9 ) log y ote: I used prentheses round the rgument on the logrithms t the left. Some tetooks my not include the prentheses. log ) log is equivlent m 000 to log m III Power Rule m Power Rule for Eponents: n mn (when power is rised to nother power, the eponents re multiplied) Likewise, when logrithm s rgument hs n eponent, the eponent is multiplied y the logrithm. p Power Rule for Logrithms: logm plogm The logrithm of power is the product of the eponent nd the logrithm. We cn lso use the power rule to epnd logrithm. (Assume ll vriles represent positive vlues.) E : Use the power rule to epnd ech logrithm nd simplify where possile. 8 ) log ) log 5(5 ) c) ln y

log ) log is equivlent m 000 to log m III Power Rule m Power Rule for Eponents: n mn (when power is rised to nother power, the eponents re multiplied) Likewise, when logrithm s rgument hs n

4 MA 5800 Lesson 6 otes Summer 06 IV Summry of properties of LOGARITHMS All vriles represent positive vlues nd ll ses re positive numers other thn. ) log 0 logrithm of is zero ) log ) log 4) log 5) log ( M) log M log Product Rule M 6) log log M log Quotient Rule p 7) log M p log M Power Rule E 4: Use the properties of logrithms to epnd ech logrithmic epression. Assume ll vriles represent positive vlues. (Epress ech in terms of logrithms of, y, nd/or z.) ) log y z 4 ) ln yz 5 4

) log 0 logrithm of is zero ) log ) log 4) log 5) log ( M) log M log Product Rule M 6) log log M log Quotient Rule p 7)

5 MA 5800 Lesson 6 otes Summer 06 c) log y d) log 7 y The reverse of epnding logrithm or logrithmic epression is condensing logrithmic epression or writing the epression s one logrithm. Agin, ssume ll vriles represent positive vlues. Hint: It is helpful to rrnge the sutrction t the end of the line nd then fctor out the negtive or sutrction sign!! E 5: Condense ech logrithm or logrithmic epression. Write s single logrithm. ) log log log y log z ) log( ) log log 4 c) (ln ln y) ln( ) 5

Hint: It is helpful to rrnge the sutrction t the end of the line nd then fctor out the negtive or sutrction sign!

6 MA 5800 Lesson 6 otes Summer 06 d) log( y ) log y log y Hint: For the emple ove, it might e wise to epnd where possile first, nd then condense into one logrithm. E 6: Use the properties of logrithms nd the informtion elow to evlute ech logrithm. log m.89, log n.89, nd log r ) log ( m n) ) log n r E 7: Use the properties of logrithms nd the logrithms elow to find the following vlues. log 8.898, log.87, nd log ) log 4 (prolem 7 continued on the net pge) 6

E 6: Use the properties of logrithms nd the informtion elow to evlute ech logrithm. log m.89, log n.89, nd log r 0.

7 MA 5800 Lesson 6 otes Summer 06 ) log 88 c) log d) log 44 E 8: Let log 4 And log 5 B. Write ech epression in terms of A nd/or B. ) log 5 ) log 4 5 E 9: Use the properties of logrithms to ssist in solving these equtions. ) log () log 5 log ) ln ln( 6) ln 9 7

) log 5 ) log 4 5 E 9: Use the properties of logrithms to ssist

8 MA 5800 Lesson 6 otes Summer 06 c) log ( ) log ( 5) d) log ( ) log 7 log ( 4) 5 log5 E 0: When the volume control on stereo system is incresed, the voltge cross loudspeker chnges from V to V, nd the deciel increse in gin is given y d 0log V. Approimte the V deciel increse if the voltge chnges from volts to 4.5 volts. 8

cross loudspeker chnges from V to V, nd the deciel increse in gin is given y d

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered:

Appendix D: Completing the Square and the Quadratic Formula. In Appendix A, two special cases of expanding brackets were considered: Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you