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Warm-up 3/13/08

Warm-up 3/13/08. Solve each equation. x 2 = 169 w 3 = 216 y 3 = -1/8 z 4 = 625. Heads Up!. Today is the last day for any make ups in the first 9 weeks

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Warm-up 3/13/08

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  1. Warm-up 3/13/08 Solve each equation. • x2 = 169 • w3= 216 • y3 = -1/8 • z4 = 625

  2. Heads Up! • Today is the last day for any make ups in the first 9 weeks • If you “borrowed” any calculators, please return them; I will be checking all of them by their serial numbers this afternoon, if you checked one out and it’s missing, you’ll be fined for it. • Book Check

  3. Copy SLM for Unit 6 (chapter 6)

  4. §6.1: nth Root Functions LEQ: How do you use rational roots to model situations? Defining the exponential function If b is any number such that b>0 and b≠0 then an exponential function is a function in the form, f(x)=bx b is the base x can be any real number.

  5. Why? • Why can’t x = 0? • Any number raised to the 0 power is 1, and that would make it a constant function • Why can’t x be negative? • If x was negative and it was raised to a fraction power, it would make the answer imaginary. • Ex) -4(1/2) = √-4 = 2i

  6. Powers and Roots • If x2 = k • X is a square root of k • If x3 = k • X is a cube root of k • If xn = k • X is an nth root of k

  7. How many real roots are there? • When “n” is odd, • K has exactly one real nth root • When “n” is even and k is positive • Ex) x2n = k • K has two real nth roots • When “n” is even and k is negative • Ex) x2n = -k • K has no real nth roots (imaginary roots)

  8. Nth root, nth power • Taking the nth root and the nth power of a number are inverse operations. • Ex) 104 = (10000)1/4 • In general: • F(x) = xn • G(x) = x1/n Are inverses of each other.

  9. Nth Root Functions • Functions with equations in the form y = x1/n, where n is an integer > 2 are called nth root functions • X1/n is defined only when x≥0 (in this chapter) • The domain of these functions is all positive real numbers • The range is also the set of all positive real numbers. • Graph some: y = x1/2; y = x1/3; y = x1/4

  10. Common Characteristics What are some common characteristics of nth root function graphs? • All of them start at (0,0) • All of them pass through (1,0)

  11. Inverse Refresh… To undo a square, you square root. To unto a cube, you cube root. To undo an exponent of 4, you take the 4th root. You can use a radical symbol or a rational exponent to indicate roots. *Different ways to use calculators…

  12. Ex. y = 5√32 index (the root you want) y = 321/5 The principal root is the positive root. Ex. 4√81 = +9, -9 +9 principal root Rational Exponent Property bm/n = n√bm

  13. 82/3 43/2 (53/4)4/3 2434/5 4 8 5 81 Simplify each expression

  14. Solving in terms of variables Express the radius r in mm of a spherical ball bearing as a function of the volume V in mm3. (Solve the equation for r) V = 4/3 πr3

  15. Refresh on Properties x5x4 x9 (xy)2 x2y2 x6÷x4 x ≠ 0 x2 (x/y)5 x ≠ 0 x5/y5 x0 1 X-n 1/(xn)

  16. Assignment Section 6.1 p. 374 – 375 #1 – 7, 10 – 12, 14, 16, 17 - 19

  17. Guided Practice 6.2 Worksheet

  18. Warm-up 3/17/08 Rewrite each expression using a radical sign. • x7/8 • x-8/5 Evaluate without a calculator. • 641/3 4) 16-3/4 5) 641/2

  19. Quick Review In textbooks, p.381 Complete #15 – 17 In notes

  20. §6.3: Converting Exponentials LEQ: How do you convert between exponential and logarithmic forms? How do you solve problems arising from exponential or logarithmic forms? Logarithms are the “opposite” of exponents; they “undo” exponentials. Logarithms have a specific relationship to exponentials…

  21. The relationship Exponential Equation: y = bx is equivalent to: Logarithmic Equation: logb(y) = x “Log-base-b of y equals x.”

  22. Quick Summary

  23. Ex. Convert 63 = 216 to the equivalent logarithmic form. Base is 6, exponent is 3… log6(216) = 3 Ex. Convert log4(1024) = 5 to the equivalent exponential expression. Base is 4, exponent is 5 45 = 1024

  24. Exponent in a different form • log443= 3 43 = 43 • log552 = x 5x = 52 • log225 = y 2y = 25 y = 5 • logbb2 = z bz = b2 z = 2

  25. From Logs to Exponents • 52 = 25 base is 5, exp is 2: log5(25) = 2 • (1/2)3 = 1/8 log(1/2)(1/8) = 3 • 24 = 64 log2(64) = 4 To evaluate logarithms, you can write them in exponential form…

  26. Evaluating Exponents Ex. Log816 8x = 16 Both can be written with a base of 2: 23x = 24 Now you can set the exponents equal (since the bases are equal) 3x = 4 Solve for x X = 4/3 So, log816 = 4/3

  27. Ex) Evaluate log5125. 5x = 125 5x = 53 Since both bases are five, you can set the exponents equal. X = 3 Thus, log5125 = 3

  28. A short cut To evaluate log216, you can ask yourself “What power of 2 is equal to 16”. • What question would you ask to evaluate log327? Evaluate it. • What question would you ask to evaluate log10100? Evaluate it.

  29. General Forms • What is the value of logb1? • Log21 c) Log31 • Log41 d) Log51 • What is the value of logbb? • log22 c) log44 • Log55 d) log99 • Explain why the base b in y = logbx cannot equal 1.

  30. Solve 10x = 4 to “undo” power of 10, log… log10x = log4 x = log4 x = 0.60206

  31. Solve to the nearest tenth log x = 3.724 103.724 = x x ≈ 5296.6 To check, Log (5296.6) = 3.724? Yes!

  32. Inverse Graphs Logarithms and exponentials are also inverses of each other. If you graph y = 10x and y = logx, they are inverses. Their graphs are reflections of each other over the line y = x. *log has a general base of 10* Find the inverse function of each: • Y = 3x 2) y = 5x 3) y = bx 1) Y = log3x 2) y = log5x 3) y = logbx

  33. Practice p. 387 - 388 #16 - 21 6.3 Worksheet Homework Section 6.3 p.387-388 #1 – 7, 9 - 15

  34. Warm-up 3/18/08 • Consider two spheres, one with a radius of 2cm and the other with twice the volume of the first. • Find the volume of the larger sphere. • Find the radius of the larger sphere. • The concentration of a hydrogen ions in a aqueous solution is given by the formula [H+] = 10(-pH) • Find the concentration if the pH is 1.5 • If the concentration is 0.00005, what is the pH?

  35. Practice p. 387 - 388 #16 - 21 6.3 Worksheet Homework Section 6.3 p.387-388 #1 – 7, 9 - 15

  36. Quiz 6.1 - 6.3 Group Quiz (You may use a partner, notes, book, etc, but keep in mind you have a test coming up soon.) It may be a good idea to use each other to “check” your work.

  37. Warm-up 3/19/08 • Use a calculator to give a 3-place decimal approximation to ln2 through ln10. • Which of them are sums of two other logarithms?

  38. Activity Class assignment p.389 #1 - 3d Discuss

  39. Assignment Read p. 390 – 394 Do: 6.4 WS

  40. §6.5: Properties of Logarithms LEQ: How do you use properties of logarithms to simplify logarithmic problems? “Recall” the properties (p.398-400)

  41. Properties of Logarithms 1) logbMN = logbM + logbN Product Property • logbM/N = logbM – logbN Quotient Property • logbMk= klogbM Power Property

  42. Examples Write the expression in single log form. • log320 – log34 log320/4 = log35 2) 3log2x + log2y log2x3 + log2y = log2x3y • 3log2 + log4 – log 16 log(23 x 4)/16 = log 32/16 = log2

  43. Examples of expansion Expand each logarithm. • log5(x/y) log5x – log5y • log3r4 log3 + logr4 = log3 + 4 log r • Can you expand log3(2x + 1)? NO, the sum can’t be factored.

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