1. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 1 Chapter 5 Logarithmic Functions

2. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 2 5.5 Using the Power Property with Exponential Models to Make Predictions

3. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 3 Example: Using the Power Property to Make a Prediction A person invests $7000 in a bank account with a yearly interest rate of 6% compounded annually. When will the balance be $10,000?

4. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 4 Solution Let B = f(t) be the balance (in thousands of dollars) after t years or any fraction thereof. We can model the situation well by using an exponential model of the form f(t) = ab t with y-intercept (0, a). The B-intercept is (0, 7). By the end of each year, the account has increased by 6% of the previous year’s balance, so b = 1.06. Thus, f(t) = 7(1.06) t

5. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 5 Solution To find when the balance is $10,000 (B = 10), substitute 10 for f(t) and solve for t:

6. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 6 Solution So, it will take about 6 years and 45 days for the balance to reach $10,000. We use a graphing calculator table to check that the input 6.1212 leads approximately to the output 10.

7. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 7 Making a Prediction To make a prediction about the independent variable t of an exponential model of the form f(t) = ab t, we substitute a value for f(t) and divide both sides of the equation by the coefficient a. Next, take the log of both sides of the equation and use the power property to help solve for t.

8. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 8 Example: Using the Power Property to Estimate Doubling Time The equation f(t) = 291(1.08) t can be used to model federal debt amounts, where f(t) is the federal debt (in billions of dollars) at t years since 1960. Estimate how often the federal debt doubles.

9. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 9 Solution In 1960, the federal debt was $291 billion. Find the year when the debt was 2(291) = 582 billion dollars (twice as large) by substituting 582 for f(t) in the equation. Calculations are shown on the next slide.

10. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 10 Solution According to the exponential function f, it took about 9 years to double the 1960 debt.

11. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 11 Half-life The half-life of an element is the amount of time it takes for the number of atoms to be reduced by half. All organisms are, in part, composed of the elements carbon-12 and carbon-14.

12. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 12 Carbon-12 and Carbon-14 Carbon-14 is radioactive. After an animal or plant dies, its carbon-14 decays exponentially with a half-life of 5730 years. However, the carbon-12 remains constant. Scientists know the ratio of carbon-14 to carbon- 12 in living organisms. Hence, scientists can determine how long ago an organism lived by measuring the decreased ratio of carbon-14 to carbon-12.

13. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 13 Example: Using the Power Property to Make and Estimate A violent volcanic eruption and subsequent collapse of the former Mount Mazama created Crater Lake, the deepest lake in the United States. Scientists found a charcoal sample from a tree that burned in the eruption. If only 39.40% of the carbon-14 remains in the sample, when did Crater Lake form?

14. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 14 Solution Let P = f(t) be the percentage of carbon-14 that remains at t years after the sample formed. Since the percentage is halved every 5730 years, we will find an exponential equation of the form f(t) = ab t.

15. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 15 Solution At time t = 0, 100% (all) of the carbon-14 remained, so the P-intercept is (0, 100). Therefore, a = 100 and f(t) = 100b t. At time t = 5730, of the carbon-14 remained. So, substitute the coordinates of the point (5730, 50) into the equation and solve for b.

16. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 16 Solution The equation is f(t) = 100(0.999879) t. We use more digits than usual for the base, as even a small change in it would greatly affect estimates well into the past (or future).

17. Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 5.5, Slide 17 To estimate the age of the sample, substitute 39.40 for f(t) and solve for t: So, the age of Crater Lake (and the sample) is approximately 7697 years. Solution