Demana Waits Foley Kennedy 3 5 Equation Solving

Demana, Waits, Foley, Kennedy 3. 5 Equation Solving and Modeling Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 1

What you’ll learn about n n n Solving Exponential Equations Solving Logarithmic Equations Orders of Magnitude and Logarithmic Models Newton’s Law of Cooling Logarithmic Re-expression … and why The Richter scale, p. H, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2

One-to-One Properties Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 3

Example: Solving an Exponential Equation Algebraically Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 4

Solution Can also use Change of Base Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 5

Example: Solving a Logarithmic Equation Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 6

Solution This uses 1: 1 property, you can also jump directly to rewriting as an exponential (line 3) Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 7

Example: Solving a Logarithmic Equation ln Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 8

Solution ln ln ln Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 9

Solution Solve ln(2 x+1) + ln(x+3) =ln(8 – 2 x) ln ln Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 10

Example: Solving a log equation n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 11

Solution n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 12

Example n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 13

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Example n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 15

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Example n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 17

Solution n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 18

Example n Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 19

Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Orders of magnitude can be used to compare any like quantities: n A kilometer is 3 orders of magnitude longer than a meter. n A dollar is 2 orders of magnitude greater than a penny. n New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 20

Richter Scale Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 21

p. H In chemistry, the acidity of a water-based solution is measured by the concentration of hydrogen ions in the solution (in moles per liter). The hydrogen-ion concentration is written [H+]. The measure of acidity used is p. H, the opposite of the common log of the hydrogen-ion concentration: p. H = –log [H+] More acidic solutions have higher hydrogen-ion concentrations and lower p. H values. Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 22

Newton’s Law of Cooling Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 23

Example: Newton’s Law of Cooling A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC? Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 24

Solution A hard-boiled egg at temperature 100ºC is placed in 15ºC water to cool. Five minutes later the temperature of the egg is 55ºC. When will the egg be 25ºC? Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 25

Regression Models Related by Logarithmic Re-Expression n n Linear regression: Natural logarithmic regression: Exponential regression: Power regression: y = ax + b y = a + blnx y = a·bx y = a·xb Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 26

Three Types of Logarithmic Re-Expression Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 27

Three Types of Logarithmic Re-Expression (cont’d) Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 28

Three Types of Logarithmic Re-Expression (cont’d) Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 29