1. Quiz 3-4 1.Convert to a common logarithm logarithm 2. Expand: 3. Condense:

2. 3.5 Equation Solving and Modeling

3. 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 examples of real world phenomena that are modeled by logarithmic and exponential functions.

4. One-to-One Properties The book teaches this “new” property. I hate new properties to remember if I don’t need them!!!! to remember if I don’t need them!!!! “undo” base ‘b’ Why memorize a new property if you can get the same result by just using properties already learned? by just using properties already learned? One-to-One property: for any exponential function: If then: u = v

5. One-to-One Properties The book teaches this “new” property. I hate new properties to remember if I don’t need them!!!! to remember if I don’t need them!!!! “undo” log base ‘b’ Why memorize a new property if you can get the same result by just using properties already learned? by just using properties already learned? If then: u = v One-to-One property: for any logarithm function:

6. Solving an Exponential Equation Algebraically Isolate the base and the exponent: “undo” base ½ Solve: Can you rewrite 1/8 as a power of ½?

7. Your Turn: solve for x 1. 2.

8. Solving a Logarithmic Equation “undo” log base 10 Solve for ‘x’

9. Your turn: 3. 4.

10. Solving a Logarithmic Equation Isolate the varable: Subtract 4 from both sides from both sides Divide both sides by 2 “undo” log base ‘e’ Solve for ‘x’

11. Your Turn: 5. 6.

12. Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. Distance: Mercury to the sun Distance: Pluto to the sun What is the order of magnitude difference when comparing the distance of Mercury and Pluto from the sun? About 2 orders of Mag.

13. Orders of Magnitude The common logarithm of a positive quantity is its order of magnitude. What is the order of magnitude difference when comparing the distance of Mercury and Pluto from the sun? Order of magnitude: common log of the number Difference in order of magnitude between two numbers: difference (subtract) the common logs of the two numbers difference (subtract) the common logs of the two numbers

14. Your Turn: 7. What is the order of magnitude difference between a gram and a kilogram? between a gram and a kilogram? 8. What is the order of magnitude difference between a person’s $100,000 salary and the $1 Trillion the a person’s $100,000 salary and the $1 Trillion the government spent that it didn’t have last year? government spent that it didn’t have last year?

15. Earthquake Intensity Is measured by the amplitude of the vibration felt at the measuring station. felt at the measuring station. T: Period of the seismic wave (seconds) B: a “fudge factor” to account for weakening of the seismic wave from origin to where it is measured. seismic wave from origin to where it is measured. Amplitude is measured in

16. Comparing Orders of Magnitude of Earthquake Intensities 2010 Gugerat, India: R = 7.9 1999 Athens, GreeceR = 5.9 We want to compare the amplitudes.

17. Comparing Orders of Magnitude of Earthquake Intensities 100 times greater !! 7.9 – 5.9 is the difference in magnitudes (difference between the natural logs of two different numbers). between the natural logs of two different numbers). The difference in magnitudes is the power of 10 difference between the two different numbers). between the two different numbers).

18. 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 + ] pH = -log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values. pH is the magitude (common log) of the hydrogen ion. The difference in pH between two solutions is the is the power of 10 difference between their hydrogen ion concentrations. of 10 difference between their hydrogen ion concentrations.

19. Your Turn: 9. The pH of carbonated water is 3.9 and the pH of household ammonia is 11.9. What are their hydrogen household ammonia is 11.9. What are their hydrogen ion concentrations? ion concentrations? 10. For problem #9, how many times greater is the hydrogen ion concetration of carbonate water that hydrogen ion concetration of carbonate water that that of ammonia? that of ammonia? pH = -log [H + ] 11. By how many orders of magnitude do the concentrations differ? differ?

20. Newton’s Law of Cooling is the temperature of the surrounding medium is the temperature of the surrounding medium is the original temperature of the object is the original temperature of the object Isaac Newton discovered that a heated object will cool at a rate Isaac Newton discovered that a heated object will cool at a rate that is dependent upon the tempurature of the surrounding that is dependent upon the tempurature of the surrounding medium. The object’s temperature can be modeled by: medium. The object’s temperature can be modeled by: is time is time is a “fudge factor” for cooling time is a “fudge factor” for cooling time

21. 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? T(5) = 55 We must first calculate “k” the “fudge factor” for this specific situation. situation. Isolate the variable

22. 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? T(5) = 55 Now we know the complete modeling equation we can find the time to reach 25 degrees  T(t) = 25 the time to reach 25 degrees  T(t) = 25 Isolate the variable

23. Your turn: A cake taken out of the oven at temperature of 350 º F. It is placed on in a room with an ambient temperature of 70 º F to cool. Ten minutes later the temperature of the cake is 150 º F. When will the cake be cool enough to put the frosting on (90 º F) ? 12. Calculate ‘k’ using the time to cool to the intermediate temperature. intermediate temperature. 13. Once you have ‘k’, find ‘t’ for T(t) = desired temperature.

24. Exponential Equations: This looks like a quadratic equation! Let:

25. Your turn: 14. Solve for ‘x’. Hint: multiply left/right by then rewrite as a “quadratic” then solve. then solve.

26. HOMEWORK Section 3-5: even problems: 2-8, 12-18, 26-36, 40-50. (20 problems)

27. Don’t worry about the following slides

28. Regression Models Related by Logarithmic Re-Expression Linear regression:y = ax + b Natural logarithmic regression:y = a + b*lnx Exponential regression:y = a·b x Power regression:y = a·x b

29. Selecting Regression Model x 1 2 3 4 5 6 y 2 5 10 17 26 38 1. Plot the data. 2.Based upon the plot, it looks like an exponential or power function exponential or power function 3.Use STAT regression feature to find the equation using each type of function. equation using each type of function. 4.Graph the curve and the data points to see how good the fit is. how good the fit is.

30. Three Types of Logarithmic Re- Expression

31. Three Types of Logarithmic Re- Expression (cont’d)