1. 4.5 Apply Properties of Logarithms p. 259 What are the three properties of logs? How do you expand a log? Why? How do you condense a log?

2. Properties of Logarithms

3. Use log 5 3≈.683 and log 5 7≈1.209 log 5 21 = log 5 (3·7)= log 5 3 + log 5 7≈.683 + 1.209 = 1.892

4. Use log 5 3≈.683 and log 5 7≈1.209 Approximate: log 5 49 = log 5 7 2 = 2 log 5 7 ≈ 2(1.209)= 2.418

5. 2. 6 log 40 = 6 log (8 5) = 8 6 log + 5 6 = 2.059 1.1610.898+ Write 40 as 8 5. Product property Simplify. ≈

6. Expanding Logarithms You can use the properties to expand logarithms. log 2 = log 2 7x 3 - log 2 y = log 2 7 + log 2 x 3 – log 2 y = log 2 7 + 3·log 2 x – log 2 y

7. Your turn! Expand: log 5mn = log 5 + log m + log n Expand: log 5 8x 3 = log 5 8 + 3·log 5 x

8. Condensing Logarithms log 6 + 2 log2 – log 3 = log 6 + log 2 2 – log 3 = log (6·2 2 ) – log 3 = log = log 8

9. SOLUTION Evaluate using common logarithms and natural logarithms. Using common logarithms: Using natural logarithms: 3 log 8 = log 8 log 3 0.9031 0.4771 1.893 3 log 8 = ln 8 ln 3 2.0794 1.0986 1.893

10. What are the three properties of logs? Product—expanded add each, Quotient— expand subtract, Power—expanded goes in front of log. How do you expand a log? Why? Use “log b ” before each addition or subtraction change. Power property will bring down exponents so you can solve for variables. How do you condense a log? Change any addition to multiplication, subtraction to division and multiplication to power. Use one “log b ”

11. For a sound with intensity I (in watts per square meter), the loudness L(I) of the sound (in decibels) is given by the function = logL(I)L(I)10 I 0 I Sound Intensity 0 I where is the intensity of a barely audible sound (about watts per square meter). An artist in a recording studio turns up the volume of a track so that the sound’s intensity doubles. By how many decibels does the loudness increase? 10 –12

12. Product property Simplify. SOLUTION Let I be the original intensity, so that 2I is the doubled intensity. Increase in loudness = L(2I) – L(I) = log10 I 0 I log10 2I2I 0 I – I 0 I 2I2I 0 I = log – = 210log I 0 I – I 0 I + ANSWER The loudness increases by about 3 decibels. 10 log 2 = 3.01 Write an expression. Substitute. Distributive property Use a calculator.

13. 4.5 Assignment page 262, 7-41 odd

14. Properties of Logarithms Day 2 What is the change of base formula? What is its purpose?

15. Your turn! Condense: log 5 7 + 3·log 5 t = log 5 7t 3 Condense: 3log 2 x – (log 2 4 + log 2 y)= log 2

16. Change of base formula: a, b, and c are positive numbers with b≠1 and c≠1. Then: log c a = log c a = (base 10) log c a = (base e)

17. Examples: Use the change of base to evaluate: log 3 7 = (base 10) log 7 ≈ log 3 1.771 (base e) ln 7 ≈ ln 3 1.771

18. Use the change-of-base formula to evaluate the logarithm. 5 log 8 SOLUTION 5 log 8 = log 8 log 5 0.9031 0.6989 1.292 8 log 14 SOLUTION 8 log 14 = log 14 log 8 1.146 0.9031 1.269

19. What is the change of base formula? What is its purpose? Lets you change on base other than 10 or e to common or natural log.

20. 4.5 Assignment Day 2 Page 262, 16- 42 even, 45-59 odd