1. CALCULUS CHAPTER 1 PT 2 Unit Question: What is a logarithmic function and how are they used to solve real-world problems?

2. SECTION 1-5 DAY 1 EQ: What is the relationship between exponential equations and logarithmic equations?

3. FUNCTIONS AND LOGARITHMS

4. EXAMPLE Given f(x) = a x find f -1 (x) y = a x x = a y How can this be solved?

5. DETERMINING IF 2 EQUATIONS ARE INVERSES

6. SYMMETRY TO Y=X  If f -1 (x) = f(x) then there is symmetry to the line y = x EX: 3x + 3y = 5equation4x – 4y = 8 3y + 3x = 5inverses4y – 4x = 8 Is there symmetry to y = x?

7. HW: Worksheet 1-5-1

8. SECTION 1-5 DAY 2 EQ: What are the Properties of Logarithms? EQ: How do you solve equations involving logarithms? EQ: What is a natural log?

9. FINDING THE INVERSE OF AN EXPONENTIAL y = 3 x x = 3 y log x = log 3 y Not as easy to solve for y when y is the exponent so we remember the primary rule of equations: whatever we do to one side we must do to the other. In this case we take the logarithm of both sides

10. PROPERTIES OF LOGARITHMS Primary Rule of Logarithms log b x = y becomes x = b y Solve: log 2 4 = x log 2 x 3 = 3 log 1000 = x

11. USING THE PRIMARY RULE: What would be true of the following and WHY???? log a x = 0 log a a = x means x =1 NOTE: Can’t take the log of a negative number i.e. in log b x = y the x can’t be negative why?

12. PROVING RULES OF LOGARITHMS let b = log a x and c = log a y convert x=a b y = a c multiply xy =a b a c xy = a b+c log a xy =log a a b+c convert log a xy = b + c substitute log a xy = log a x + log a y

13. ADDITIONAL RULES

14. EXAMPLES log (x 2 + 1) – log (x – 2) = 1 log (4x -4) log x =2

15. SOLVE USING THE RULES OF LOGARITHMS log 50 + log 2=xlog x = log 12 – log 3log 8 – log x = 2

16. CHANGE OF BASE FORMULA

17. SOLVE: LOGARITHMS AND NATURAL LOGS 3=4 x log 3=log 4 x log 3 = x log 4 ln 3=ln 4 x ln 3 = x ln 4

18. FINAL RULES

19. HW: Worksheet 1-5-2

20. SECTION 1-5 DAY 3 EQ: What are real-world applications of exponential and logarithmic functions?

21. OTHER FORMULAS  I = Prt   A= final amountI = interest  P = principalP = principal  r = rate as a decimalr = rate as a decimal  n = number of times compounded in one year t = time in years  t = the time in years  How are they the same and how are they different:

22. EXAMPLES  In 1900, the population of the U.S. was 3,465,000 with an annual growth rate of 6.2%. How long will it be until the population reaches 10,000,000?

23. EXAMPLES  A certain bacteria colony has a growth rate of 26% per hour. If there were 42 bacteria in the colony when the study began, how long will it take to have 258 bacteria?

24. HEADING  In 2000, the population of a county in Southeastern PA was 5,263,126. The population of this area has been decreasing at a rate of 3% per year, if this continues, when will the population go below 4,500,000?

25. EXAMPLES Knowing P = P 0 e rt The approximate population of Dallas was 680,000 in 1960. In 1980 it was 905,000. Find r in the growth formula and use it to approximate the population in 2010.

26. HW: Pg. 40 46 to 49

27. SECTION 1-4 DAY 1 EQ: What are parametric equations and how are they used?

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37. HW: Pg. 39