Logarithm Rules: Product, Quotient, and Power Rule with Examples, Slides of Algebra

Download Logarithm Rules: Product, Quotient, and Power Rule with Examples and more Slides Algebra in PDF only on Docsity! §9.4a Logarithm Rules Docsity.com Review §  Any QUESTIONS About • §9.3 → Common & Natural Logs  Any QUESTIONS About HomeWork • §9.3 → HW-45 9.3 MTH 55 Docsity.com Power Rule for Logarithms • Let M, N, and a be positive real numbers with a ≠ 1, and let r be any real number. Then the POWER Rule  That is, The logarithm of a number to the power r is r times the logarithm of the number. loga M r = r loga M Docsity.com Example  Product Rule • Express as an equivalent expression that is a single logarithm: log3(9∙27) • Solution log3(9·27) = log39 + log327. • As a Check note that log3(9·27) = log3243 = 5 35 = 243 • And that log39 + log327 = 2 + 3 = 5. 32 = 9 and 33 = 27 Docsity.com Example  Product Rule • Express as an equivalent expression that is a single logarithm: loga6 + loga7 • Solution = loga(42). Using the product rule for logarithms loga6 + loga7 = loga(6·7) Docsity.com Example  Power Rule • Use the power rule to write an equivalent expression that is a product: a) loga6−3 4b) log .x  Solution = log4x1/2 Using the power rule for logarithms a) loga6−3 = −3loga6 4b) log x = ½ log4x Using the power rule for logarithms Docsity.com Example  Use The Rules • Given that log5z = 3 and log5y = 2, evaluate each expression. a. log5 yz( ) b. log5 125y7( ) c. log5 z y d. log5 z 1 30 y5     a. log5 yz( )= log5 y + log5 z = 2 + 3 = 5  Solution Docsity.com Example  Use The Rules • Solution  Soln c. log5 z y = log5 z y     1 2 = 1 2 log5 z − log5 y( ) = 1 2 3− 2( )= 1 2 b. log5 125y 7( )= log5 125 + log5 y7 = log5 5 3 + 7 log5 y = 3+ 7 2( )= 17 Docsity.com Example  Use The Rules • Soln b) 1/ 3 3 7 7 log logb b xy xy z z  =     7 1 log 3 b xy z = ⋅ ( )71 log log3 b bxy z= − ( )1 log log 7log 3 b b b x y z= + − Docsity.com Caveat on Log Rules • Because the product and quotient rules replace one term with two, it is often best to use the rules within parentheses, as in the previous example ( )1 log log 7log 3 b b b x y z= + − 7 1 log 3 b xy z = ⋅ Docsity.com Example  Expand by Log Rules • Write the expressions in expanded form a. log2 x2 x −1( )3 2x +1( )4 b. logc x 3y2z5  Solution a) log2 x2 x −1( )3 2x +1( )4 = log2 x 2 x −1( )3 − log2 2x +1( ) 4 = log2 x 2 + log2 x −1( ) 3 − log2 2x +1( ) 4 = 2 log2 x + 3log2 x −1( )− 4 log2 2x +1( ) Docsity.com Example  Condense Logs • Solution a) log 3x − log 4y = log 3x 4y      Solution b) 2 log x + 1 2 ln x2 +1( )= ln x2 + ln x2 +1( ) 1 2 = ln x2 x2 +1( ) Docsity.com Example  Condense Logs • Solution c) 2 log2 5 + log2 9 − log2 75 = log2 5 2 + log2 9 − log2 75 = log2 25 ⋅9( )− log2 75 = log2 25 ⋅9 75 = log2 3 Docsity.com Example  Condense Logs • Solution d) 1 3 ln x + ln x +1( )− ln x2 +1( )  = 1 3 ln x x +1( )− ln x2 +1( )  = 1 3 ln x x +1( ) x2 +1     = ln x x +1( ) x2 +1 3 Docsity.com Summary of Log Rules • For any positive numbers M, N, and a with a ≠ 1 log log log ;a a a M M N N = − log log ;pa aM p M= log .ka a k= log ( ) log log ;a a aMN M N= + Docsity.com Typical Log-Confusion • Beware that Logs do NOT behave Algebraically. In General: loglog , log a a a MM N N ≠ log ( ) (log )(log ),a a aMN M N≠ log ( ) log log ,a a aM N M N+ ≠ + log ( ) log log .a a aM N M N− ≠ − Docsity.com WhiteBoard Work  Problems From §9.4 Exercise Set • 24, 30, 36, 58, 60  Condense Logarithm Docsity.com