Precal hw

just the ones circled

9:28 .1 5G 458 of 1094 446 EXPONENTIAL AND LOGARITHMIC FUNCTIONS In Exercises 30 - 33, use the appropriate change of base formula to convert the given expression to an expression with the indicated base. 30. 7%-1 to base e 31 log3(x + 2) to base 10 32 (3) to base e 33. log(x2 + 1) to base e In Exercises 34 - 39, use the appropriate change of base formula to approximate the logarithm. 34. log3 (12) 35. logs (80) 36. log6 (72) 37. log4 TO 38. log3 (1000) 39. log? (50) 40 Compare and contrast the graphs of y = In(x2) and y = 2 In(x). 41. Prove the Quotient Rule and Power Rule for Logarithms. 42. Give numerical examples to show that, in general, a) logo(x + y) # logb(x) + log.(y) (b) logi(x - y) # logi(x) - log (y) (c) log () + 1ogb () logo ( y ) 43. The Henderson-Hasselbalch Equation: Suppose H A represents a weak acid. Then we have a reversible chemical reaction HAHA. The acid disassociation constant, Ka, is given by K - HA] +[A-] = [H+] [HA] [HA] where the square brackets denote the concentrations just as they did in Exercise 77 in Section 6.1. The symbol pka is defined similarly to pH in that pha = - log(Ka). Using the definition of PH from Exercise 77 and the properties of logarithms, derive the Henderson-Hasselbalch Equation which states [A-] PH = pKa + 109 (H A] 44. Research the history of logarithms including the origin of the word 'logarithm' itself. Why is he abbreviation of natural log 'In' and not 'nl'? 45. There is a scene in the movie 'Apollo 13' in which several people at Mission Control use slide rules to verify a computation. Was that scene accurate? Look for other pop culture references to logarithms and slide rules. 6.2 PROPERTIES OF LOGARITHMS 447 6.2.2 ANSWERS 1. 3 In(x) + 2 In(y) 2. 7 - 10g2(22 + 4) 3. 3log5 (z) - 6 4. log (1.23) + 37 5. { In(z) - In(x) - In(y) 6. logs(x - 5) + logs(x + 5) stitz-zeager.com9:28 .1 5G Student 5Which moone if it is lying to us about the first answer it gave us, at least it is being consistent. 457 of 1094 6.2 PROPERTIES OF LOGARITHMS 445 6.2.1 EXERCISES In Exercises 1 - 15, expand the given logarithm and simplify. Assume when necessary that all quantities represent positive real numbers. 1. In(x3 y2) 2. log2 128 72 + 4 3. logs (25) 4. log(1.23 x 1037) 5. In VE 6. logs (a2 - 25) 7. log /2 ( 42 3 ) 8. log: (9x(y3 -8)) 9. log (10007345) 10. 10g3 (216) 4 8ly4 11. In (Nez) 12. log6 18 10g 100xVy 14. log 4V12 yV z 15. In 10 Vyz In Exercises 16 - 29, use the properties of logarithms to write the expression as a single logarithm. 16. 4 In(x) + 2 In(y) 17. log2(x) + log2(y) - log2(z) 18. log3(x) - 210g3(y) 19. log3(2) - 210g3(y) - 10g3(z) 21. log(x) - log(2) + log(y) 22. - 3 In(x) - 3 In(y) + 3 In(2) 23. logs(x) - 3 24. 3 - log(x) 25 log7(x) + log7(x - 3) - 2 26. In(2) + 2 log2(2) + log4(2) 28. log2(x) + log,(x - 1) 29. log2(x) + log: (x - 1) 446 EXPONENTIAL AND LOGARITHMIC FUNCTIONS In Exercises 30 - 33, use the appropriate change of base formula to convert the given expression to an expression with the indicated base. 30. 7%-1 to base e 31. log3(x + 2) to base 10 32. (2 to base e 33. log(x2 + 1) to base e stitz-zeager.com