1. Quiz 7-4: Convert to exponential form 1. 2. Convert to logarithm form logarithm form 3. 4. 5. 6. Simplify: Find the Inverse: 7. 8.

2. 7-5 Properties of Logarithmic Functions

3. What you’ll learn about Properties of Logarithms Change of Base (review) Graphs of Logarithmic Functions with Base b Re-expressing Data … and why Logarithms are used extensively in science. You need to understand their many special properties, so that you can solve problems in the real world.

4. Properties of Logarithms: This is the same thing as saying: “When you multiply “like” bases of powers, you just add the exponents.” “like” bases of powers, you just add the exponents.” Logarithmic version of the Exponent property: Product of Powers. Product of Powers. Product Rule:

5. 1. Take your pick Your Turn: “Expand the Product” 2. “Expanding the Product”

6. Your Turn: “Condense the Product” 3. 4. “Condense the Product”

7. Properties of Logarithms Quotient Rule “expand the quotient” “condense the quotient” Logarithm: another way of writing the exponent

8. Your Turn: 5. 6. Condense the quotient Expand the Quotient 8. 7.

9. Properties of Logarithms Power Rule c  Combination of expanding & simplifying

10. Your Turn: 9. 10. 11. Think of it like this:

11. Expanding the Logarithm of a Product Use the properties of logarithms to write the following as a sum of logarithms or multiple logarithms. following as a sum of logarithms or multiple logarithms.

12. Your Turn: Expand these logarithms: 12. 13.

13. Condensing a Logarithmic Expression Write the following expression as a single logarithm:

14. Your Turn: Condense the logarithm: 14.

15. Change-of-Base Formula for Logarithms Change of Base Formula: This is most often used to convert from other base logs to log base 10 (since that is other base logs to log base 10 (since that is what you calculator has on it.) what you calculator has on it.) Convert to base 10.

16. Another Example: Calculate the following using the base conversion formula. base conversion formula.

17. Your turn: Calculate the following using the base conversion formula. base conversion formula. 15. 16.

18. Approximating Expressions If you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms: values and properties of logarithms to find other logarithms: Use these to find: Use your calculator to find: log 2 = ?

19. Approximating Expressions If you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms: values and properties of logarithms to find other logarithms: Use these to find: Use your calculator to find: log 1/2 = ?

20. Approximating Expressions If you don’t have your calculator, you can use given values and properties of logarithms to find other logarithms: values and properties of logarithms to find other logarithms: Use these to find: Use your calculator to find: log 18 = ?

21. Earthquake Intensity Is measured by the amplitude of the vibration felt at the measuring station. felt at the measuring station. B: a “fudge factor” to account for weakening of the seismic wave from origin to pt of of the seismic wave from origin to pt of measurement measurement Amplitude is measured in

22. Earthquake Magnitude

23. 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 negative of the common log of the hydrogen-ion concentration: pH = -log [H + ] More acidic solutions have higher hydrogen-ion concentrations and lower pH values.

24. Newton’s Law of Cooling A high temperature item will cool off in a lower temperature medium in which it is placed. This cooling off process can be modeled by the following equation. Temperature (as a function of time) Temperature of the medium the medium Initial Temp of the object of the object Constant, determined by the heat transfer by the heat transfer characteristics of the material characteristics of the material Time

25. 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. K = 0.15 When will the egg be 25 º C? “isolate the power” “undo the base”

26. Barking dog sound intensity: watts/sq meter Sound Intensity Loudness of the sound (in decibels) as a function (in decibels) as a function of the sound intensity of the sound intensity Intensity of the sound in the sound in watts/sq meter watts/sq meter Intensity of sound at the threshold at the threshold of hearing ( of hearing ( watts per sq meter) watts per sq meter) How Loud is a dog’s bark?

27. HOMEWORK Section 7-5 Page 510: (evens) 4-42, 46-52, 76, 78 (evens) 4-42, 46-52, 76, 78 (26 points)