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Exponentials, Logarithms, and Inverses

Exponentials, Logarithms, and Inverses. Exponential Functions. Logarithm Functions. Hodgepodge. Inverses (Algebra). Inverses (Graphs). 100. 100. 100. 100. 100. 200. 200. 200. 200. 200. 300. 300. 300. 300. 300. 400. 400. 400. 400. 400. 500. 500. 500. 500. 500.

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Exponentials, Logarithms, and Inverses

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  1. Exponentials, Logarithms, and Inverses

  2. Exponential Functions Logarithm Functions Hodgepodge Inverses (Algebra) Inverses (Graphs) 100 100 100 100 100 200 200 200 200 200 300 300 300 300 300 400 400 400 400 400 500 500 500 500 500

  3. Exponentials – 100 Points • QUESTION: • (30 sec.) Simplify • ANSWER: • 7v5/2

  4. Exponentials – 200 Points • QUESTION: • (1 min) Convert to a logarithm: 36=729 • ANSWER: • Log3729 = 6

  5. Exponentials – 300 Points • QUESTION: • (1 min) Simplify the following: • 3x/3y 3x•3y (3x)y • ANSWER: • 3x-y 3x+y 3x•y

  6. Exponentials – 400 Points • QUESTION: • (2 min) Solve: 5(67x)=13 • ANSWER: • x =0.076183

  7. Exponentials – 500 Points • QUESTION: • (4 min) Solve: e2x – 6ex +8 = 0 (Hint: factor) • ANSWER: • ex = 2  x = 0.69315 • ex = 4  x = 1.38629

  8. Logarithms– 100 Points • QUESTION: • (2 min) Expand the following: Log2 √(2x(x2+2)) • ANSWER: • ½ • [Log22+Log2x+Log2(x2+2)]

  9. Logarithms – 200 Points • QUESTION: • (2 min) Write the expression as a single logarithm • 3 ln 2 – ½ ln x + ln (x+1) • ANSWER: • ln [ (8•(x+1) ) / ( √(x) ) ]

  10. Logarithms – 300 Points • QUESTION: • List the Product, Quotient and Power of a Power Logarithm properties. (1 min) • ANSWER: • Product: Logb(mn) = Logbm + Logbn • Quotient: Logb(m/n) = Logbm – Logbn • Power: Logbmn = n•Logbm

  11. Logarithms – 400 Points • QUESTION: • (1 min)What is the change of base formula • ANSWER: • Logba = Log (a) / Log (b)

  12. Logarithms – 500 Points • QUESTION: • (4 minutes) Solve for x: • ANSWER: • log3 (x+9) = log3((x-3)/(x+2)) • x+9=(x-3)/(x+2) • (x+9)(x+2) = x-3 • x2+11x+18= x-3 • x2+10x+21=0 • (x+3) (x+7) = 0 • x = -3, -7

  13. Hodgepodge – 100 Points • QUESTION: • (2 min) Expand: Log ( (7xy5) / (√(x-2) ) ) • ANSWER: • Log 7 + Log x + 5•Log y + ½ Log (x-2)

  14. Hodgepodge – 200 Points • QUESTION: • (1 min) What is the base of our general Log? What about ln? • ANSWER: • Log is base 10 unless specified • Ln is the natural logarithm with a base of e

  15. Hodgepodge – 300 Points • QUESTION: • Explain using properties why taking the natural log of e7x gives you 7x • ANSWER: • Ln (e7x) = 7x •Log (e) = 7x • 1 = 7x

  16. Hodgepodge – 400 Points • QUESTION: • What mathematician was born on Pi Day? • ANSWER: • Albert Einstein

  17. Hodgepodge – 500 Points • QUESTION: • Simplify and solve • ANSWER: • x = ±1 does -1 make sense?

  18. Inverses (Algebra) – 100 Points • QUESTION: • Find the inverse of f(x)=3x+2. • ANSWER: • f -1(x)=(x-2)/3

  19. Inverses (Algebra) – 200 Points • QUESTION: • If regular gasoline is selling for $3.95 per gallon, the price of any particular purchase p is a function of the number of gallons g in that purchase. • Write this as a function and find its inverse • ANSWER: • Function: p=3.95•g • Inverse: g=p/3.95

  20. Inverses (Algebra) – 300 Points • QUESTION: • ANSWER: • y=(7/x)-4

  21. Inverses (Algebra) – 400 Points • QUESTION: • Write out the steps for either of the two algebra styles we have learned to determine the inverse of a function • ANSWER: • Answers may vary

  22. Inverses (Algebra) – 500 Points • QUESTION: • How could you use composition of functions to check that f(x) and g(x) are inverses of each other? • ANSWER: • Test f(g(x)) and/or g(f(x)) and check that they equal x.

  23. Inverses (Graph) – 100 Points • QUESTION: • What is the line of symmetry for functions and their inverse? • ANSWER: • y=x

  24. Inverses (Graph) – 200 Points • QUESTION: • Graph the following function then graph its inverse: • y=3x+1 • ANSWER: • Inverse equation: y=(x-1)/3

  25. Inverses (Graph) – 300 Points • QUESTION: • Given the following functions, which have inverses that are functions? If it does not have one explain why. • a) b) c) • ANSWER: • a & b have inverse functions, c does not because it fails the “horizontal line test”

  26. Inverses (Graph) – 400 Points • QUESTION: • How do the domain and range of a function relate to the domain and range of its inverse • ANSWER: • They are switched, why?

  27. Inverses (Graph) – 500 Points • QUESTION: • How could I use a graph to check that my functions are inverses of each other? • ANSWER: • y=x is the line of symmetry, this is made clear when the coordinates flip (x,y)  (y,x)

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