Copyright © 2011 Pearson, Inc. 3.5 Equation Solving and Modeling.

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Copyright © 2011 Pearson, Inc. 3.5 Equation Solving and Modeling

Copyright © 2011 Pearson, Inc. Slide What you’ll learn about 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, pH, and Newton’s Law of Cooling, are among the most important uses of logarithmic and exponential functions.

Copyright © 2011 Pearson, Inc. Slide One-to-One Properties

Copyright © 2011 Pearson, Inc. Slide Example Solving an Exponential Equation Algebraically

Copyright © 2011 Pearson, Inc. Slide Example Solving an Exponential Equation Algebraically

Copyright © 2011 Pearson, Inc. Slide Example Solving a Logarithmic Equation

Copyright © 2011 Pearson, Inc. Slide Example Solving a Logarithmic Equation

Copyright © 2011 Pearson, Inc. Slide Example Solving a Logarithmic Equation

Copyright © 2011 Pearson, Inc. Slide Example Solving a Logarithmic Equation

Copyright © 2011 Pearson, Inc. Slide Example Solving a Logarithmic Equation

Copyright © 2011 Pearson, Inc. Slide 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: A kilometer is 3 orders of magnitude longer than a meter. A dollar is 2 orders of magnitude greater than a penny. New York City with 8 million people is 6 orders of magnitude bigger than Earmuff Junction with a population of 8.

Copyright © 2011 Pearson, Inc. Slide Richter Scale

Copyright © 2011 Pearson, Inc. Slide pH 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 pH, the opposite of the common log of the hydrogen-ion concentration: pH = –log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

Copyright © 2011 Pearson, Inc. Slide Newton’s Law of Cooling

Copyright © 2011 Pearson, Inc. Slide Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

Copyright © 2011 Pearson, Inc. Slide Example Newton’s Law of Cooling A hard-boiled egg at temperature 100 º C is placed in 15 º C water to cool. Five minutes later the temperature of the egg is 55 º C. When will the egg be 25 º C?

Copyright © 2011 Pearson, Inc. Slide Regression Models Related by Logarithmic Re-Expression Linear regression:y = ax + b Natural logarithmic regression:y = a + blnx Exponential regression:y = a·b x Power regression:y = a·x b

Copyright © 2011 Pearson, Inc. Slide Three Types of Logarithmic Re-Expression

Copyright © 2011 Pearson, Inc. Slide Three Types of Logarithmic Re-Expression (cont’d)

Copyright © 2011 Pearson, Inc. Slide Three Types of Logarithmic Re-Expression (cont’d)

Copyright © 2011 Pearson, Inc. Slide Quick Review

Copyright © 2011 Pearson, Inc. Slide Quick Review Solutions