4.5 Apply Properties of Logarithms

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 log53≈.683 and log57≈1.209 • log521 = • log5(3·7)= • log53 + log57≈ • .683 + 1.209 = • 1.892

4. Use log53≈.683 and log57≈1.209 • Approximate: • log549 = • log572 = • 2 log57 ≈ • 2(1.209)= • 2.418

5. log log log log 6 6 6 6 2. (8 • 5) 40 = 8 5 = + 1.161 + 0.898 2.059 = Write 40 as 8 • 5. Product property ≈ Simplify.

6. Expanding Logarithms • You can use the properties to expand logarithms. • log2 = • log27x3 - log2y = • log27 + log2x3 – log2y = • log27 + 3·log2x – log2y

7. Your turn! • Expand: • log 5mn= • log 5 + logm + logn • Expand: • log58x3 = • log58 + 3·log5x

8. Condensing Logarithms • log 6 + 2 log2 – log 3 = • log 6 + log 22 – log 3 = • log (6·22) – log 3 = • log = • log 8

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

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 “logb” 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 “logb”

11. Sound Intensity I I I L(I) 10 log = 0 0 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 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

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

13. 4.5 Assignment page 262, 7-41 odd

14. Properties of LogarithmsDay 2 • What is the change of base formula? • What is its purpose?

15. Your turn! • Condense: • log57 + 3·log5t = • log57t3 • Condense: • 3log2x – (log24 + log2y)= • log2

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

17. Examples: • Use the change of base to evaluate: • log37 = • (base 10) • log 7 ≈ • log 3 • 1.771 • (base e) • ln 7≈ • ln 3 • 1.771

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

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