Section5.4 Properties of Logarithmic Functions - PowerPoint PPT Presentation

Section5.4 Properties of Logarithmic Functions

PropertiesofLogarithms

Formulas Basic Properties: log a 1 = 0

Formulas Basic Properties: log a 1 = 0 log a a = 1

Formulas Basic Properties: log a 1 = 0 log a a = 1 log a a x = x

Formulas Basic Properties: log a 1 = 0 log a a = 1 log a a x = x a log a x = x

Formulas The Product Rule: Basic Properties: log a ( xy ) = log a x + log a y log a 1 = 0 log a a = 1 log a a x = x a log a x = x

Formulas The Product Rule: Basic Properties: log a ( xy ) = log a x + log a y log a 1 = 0 The Quotient Rule: log a a = 1 � � log a = log a x − log a y x log a a x = x y a log a x = x

Formulas The Product Rule: Basic Properties: log a ( xy ) = log a x + log a y log a 1 = 0 The Quotient Rule: log a a = 1 � � log a = log a x − log a y x log a a x = x y The Power Rule: a log a x = x log a ( x z ) = z log a x

Examples Calculate the following: 1. log 4 64

Examples Calculate the following: 1. log 4 64 3

Examples Calculate the following: 1. log 4 64 3 2. 10 log 51

Examples Calculate the following: 1. log 4 64 3 2. 10 log 51 51

Examples Calculate the following: √ 3. log 3 27 1. log 4 64 3 2. 10 log 51 51

Examples Calculate the following: √ 3. log 3 27 1. log 4 64 3 3 2 2. 10 log 51 51

Examples Calculate the following: √ 3. log 3 27 1. log 4 64 3 3 2 2. 10 log 51 4. ln e 200 51

Examples Calculate the following: √ 3. log 3 27 1. log 4 64 3 3 2 2. 10 log 51 4. ln e 200 51 200

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10 1 + 4 log 2 x − 2 log 2 y − 10 log 2 z

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10 1 + 4 log 2 x − 2 log 2 y − 10 log 2 z √ 3 x 2 − 2 x 6. log 4

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10 1 + 4 log 2 x − 2 log 2 y − 10 log 2 z √ 3 x 2 − 2 x 6. log 4 1 3 log 4 x + 1 3 log 4 ( x − 2)

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10 1 + 4 log 2 x − 2 log 2 y − 10 log 2 z √ 3 x 2 − 2 x 6. log 4 1 3 log 4 x + 1 3 log 4 ( x − 2) � 3 x 7. log 5 y

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10 1 + 4 log 2 x − 2 log 2 y − 10 log 2 z √ 3 x 2 − 2 x 6. log 4 1 3 log 4 x + 1 3 log 4 ( x − 2) � 3 x 7. log 5 y 1 2 log 5 3 + 1 2 log 5 x − 1 2 log 5 y

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10 1 + 4 log 2 x − 2 log 2 y − 10 log 2 z √ 3 x 2 − 2 x 6. log 4 1 3 log 4 x + 1 3 log 4 ( x − 2) � 3 x 7. log 5 y 1 2 log 5 3 + 1 2 log 5 x − 1 2 log 5 y 8. Given that log a 2 ≈ 0 . 301 and log a 7 ≈ 0 . 845, find log a 28.

Examples (continued) Use the Properties of Logarithms to expand the expression into sums and differences of logarithms: 2 x 4 5. log 2 y 2 z 10 1 + 4 log 2 x − 2 log 2 y − 10 log 2 z √ 3 x 2 − 2 x 6. log 4 3 log 4 x + 1 1 3 log 4 ( x − 2) � 3 x 7. log 5 y 1 2 log 5 3 + 1 2 log 5 x − 1 2 log 5 y 8. Given that log a 2 ≈ 0 . 301 and log a 7 ≈ 0 . 845, find log a 28. 1.447

Examples (continued) Use the Properties of Logarithms to combine the expression into a single logarithm: 9. − 2 ln 3 + 4 ln x − 1 2 ln( x − 1)

Examples (continued) Use the Properties of Logarithms to combine the expression into a single logarithm: 9. − 2 ln 3 + 4 ln x − 1 2 ln( x − 1) x 4 ln 9 √ x − 1

Examples (continued) Use the Properties of Logarithms to combine the expression into a single logarithm: 9. − 2 ln 3 + 4 ln x − 1 2 ln( x − 1) x 4 ln 9 √ x − 1 10. log 4900 − 2 log 7

Examples (continued) Use the Properties of Logarithms to combine the expression into a single logarithm: 9. − 2 ln 3 + 4 ln x − 1 2 ln( x − 1) x 4 ln 9 √ x − 1 10. log 4900 − 2 log 7 2

Examples (continued) Use the Properties of Logarithms to combine the expression into a single logarithm: 9. − 2 ln 3 + 4 ln x − 1 2 ln( x − 1) x 4 ln 9 √ x − 1 10. log 4900 − 2 log 7 2 11. 4 log 5 ( x − 4) + log 5 ( x − 1) − 3 log 5 ( x 2 − 5 x + 4)

Examples (continued) Use the Properties of Logarithms to combine the expression into a single logarithm: 9. − 2 ln 3 + 4 ln x − 1 2 ln( x − 1) x 4 ln 9 √ x − 1 10. log 4900 − 2 log 7 2 11. 4 log 5 ( x − 4) + log 5 ( x − 1) − 3 log 5 ( x 2 − 5 x + 4) x − 4 log 5 ( x − 1) 2