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Precalculus Review Worksheet for second Chapter 4 test _____________ name For the test, you must know the exponential growth equations. They will not be provided for you. Very specific formulas, like Newton’s law of cooling, will be provided. t a. A (t) = A0 (1 + r)t c. A (t) = A0 ert b. A (t) = A0 b k You also must know the laws of logarithms. They will not be provided to you. Remember M d. log b MN = log b M + log b N e. log b = log b M log b N f. log b Mk = k log b M N logbx = a IFF ba = x log c ln c log a c g. log bc = log b ln b log a b The test will have 2 pages. On the first page, you will not be able to use a calculator. On the second, you will. I. Simplify the following expressions, if possible. Answers to log problems should be exact - either in fractional or radical form. Get the answer w/out a calculator! 1. log 1000 2. log .001 6. ln e2 10. 7. ln 1 ln 25 ln 2 2 14. ln e3 19. log3 3 81 3. log 3 81 1 8. e3 4. log 5 .04 9. log 3 + log 6 log 2 log 5 2 + log 5 3 11. 2 log P log Q 12. log b M + log b N 3 log b P 15. ln e5y 16. 10log 6 20. log(log 100100) *21. log4(log228) 17. 10 2+logx *22. 5. log 7 ( 7 3 7 ) log 4 32 log 8 16 13. ln c + 1 ln d 3 18. 10 4log2 *23. If logbM = 3, then logMb = ? II. Solve for x. Try not to use a calculator. Leave answers in exact form. 24. log x = 5 25. x3 = 20 26. log | x | = 2 27. ln ( ln x ) = 2 28. log 5 ( log 4 x ) = 0 29. log 3 ( log 5 x ) = 1 30. (log x)3 = 27 31. 3x+1 = 62x (*problems taken from math team competitions) 32. log 5 x + log 5(x 4) = 1 33. log 3( x + 1 ) + log 312 = 4 34. log 9 (x2 + 12) = log 9 x + log 9 7 35. log (3x 1) log (2x + 5) = 2 36. 32x 11 3x 12 = 0 37. 38. 3x = 26 39. e x = 70 e2x 8ex + 15 = 0 40. 52x = 100 III. Solve the following word problems 41. An investment is made at 5.75% compounded quarterly. How long does it take to double the investment? 42. If an investment of $1000 is compounded continuously, what interest rate must you get to triple the money in 12 years? (round to hundredth of a percent) 43. Uranium 238 has a half life of 4.5 billion years (!!). How long will it take 1 kg to decay to 1 g? 44. How long does it take $500 to become $2000 if it is invested at 3.25% and compounded a) yearly b) continuously ? 45. Decibel Level is found by using this formula dB = 10 log I . I0 What is the decibel level of 3 stereos, each playing at 72 dB? 46. The population of a certain country was 2,345,235 in 1985 and 3,458,377 in 1995. Assume that the growth is exponential. (a) Find a formula for the population t years after 1985. (b) Find the time required for the population to double. (b) Use the data to predict the population in the year 2005. 47. A kettle full of water is brought to a boil in a room with temperature 32° C. After 5 minutes the temperature of the water has decreased from 100° C to 80°C. Find the temperature after another 5 minutes. 48. A substance has a hydrogen ion concentration of [ H+ ] = 2.1 * 10-2 M. Find the pH of the substance.