Example 1: Express as a sum of logarithms by using the Product Rule. (By the definition of log)
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1 Section 5. Properties of Logrithmic Functions Section 5. Properties of Logrithmic Functions This section covers some properties of rithmic function tht re very similr to the rules for exponents. Properties of Logrithms For ny positive number M nd N, nd ny rithmic bse, Product Rule: ( ) M N M + N Quotient Rule: M N M Product Rule: p M p M N Exmple : Express s sum of rithms by using the Product Rule. ( ) 97 (By the definition of ) Exmple : Express s single rithm. p + q Exmple : Express s product. Compre this to the left side of the Power Rule: M nd p p M p M. Now inserting these in to the right side of the power rule gives. Express 7 s product. First rewrite 7 s n exponent lo g 7. Then use the Power Rule: l og 7. using n x x n : Copyright 06 Person Eduction, Inc.
2 Chpter 5 Exponentil Functions nd Logrithmic Functions 6 Express ln x s product. Using the Power Rule: ln 6 x. Exmple : Express s difference of rithms 8 t ( By the Quotient Rule) w Exmple 5: Express s single rithm b 6 b 6 (Simplifying the frction) Exmple 6: 5 Express in terms of sums nd differences of rithms 5 Express b c in terms of sums nd differences of rithms 5 5 n n (Rewrite s n exponent using x x ) (Distributing) Copyright 06 Person Eduction, Inc.
3 Section 5. Properties of Logrithmic Functions 5 y Express b in terms of sums nd differences of rithms mn 5 (Distributing) Exmple 7: Express s single rithm 5b x b y+ b (By the Product Rul e) Exmple 8: Express s single rithm ln x+ ln x 5x ( ) ( ) (By fctoring the denomintor) (By cnceling x + ) These properties of rithms cn lso be used to find some unknown rithm when given some prticulr rithmic vlues. Exmple 9: Given tht 0.0 nd 0.77, find 6 Copyright 06 Person Eduction, Inc.
4 Chpter 5 Exponentil Functions nd Logrithmic Functions 6 (Rewriting 6 s ) (Substituting in the given vlues of nd (Adding) ) Given tht 0.0nd 0.77, find (Substituting in the given vlues of nd ) (Subtrcting) Given tht 0.0 nd 0.77, find 8 8 (By noticing tht 8 ) (Substituting in the given vlues of ) (Multiplying) Given tht 0.0nd 0.77, find (Using the Quotient Rule) (Since 0, nd by noticing tht ) (Using the Power Rule) (Substituting in the given vlues of ) (Multiplying) Given tht 0.0 nd 0.77, find 5 Copyright 06 Person Eduction, Inc.
5 Section 5. Properties of Logrithmic Functions 5 5 (Writing 5 s + ) However, we cnnot rewrite this using ny of our properties. Given tht 0.0nd 0.77, find (Substituting in the given vlues of nd ) (Dividing) Another useful properties for simplifying rithms is given below. The Logrithm of Bse to Power For ny bse nd ny positive rel number x x x Exmple 0: 8 x (By the property x) 8 ln e (Rewriting ln s e ) x (By the property x) k 0 (Rewriting s ) 0 x (By the property x ) A Bse to Logrithmic Power For ny bse nd ny positive rel number x x x Copyright 06 Person Eduction, Inc.
6 6 Chpter 5 Exponentil Functions nd Logrithmic Functions Exmple : k (By the property x x ) e ln5 (Rewriting ln s ) e (By the property x x) 7t 0 (Rewriting s ) 0 (By the property x x) Copyright 06 Person Eduction, Inc.
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