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Exponential/ Logarithmic

Exponential/ Logarithmic. Exponential Functions. f(x) = a x Domain (- ∞, ∞) Range (0, ∞) Three types: 1) if 0 < a < 1 2) if a = 1 3) if a > 1. Laws of Exponents. a x + y = a x a y a x / a y = a x –y (a x ) y = a xy (ab) x = a x b x. Sketching Example.

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Exponential/ Logarithmic

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  1. Exponential/ Logarithmic

  2. Exponential Functions f(x) = ax Domain (-∞, ∞) Range (0, ∞) Three types: 1) if 0 < a < 1 2) if a = 1 3) if a > 1

  3. Laws of Exponents a x + y = ax ay ax/ ay = a x –y (ax)y = axy (ab)x = axbx

  4. Sketching Example Sketch the function y = 3 – 2x

  5. Exponential Functions are One to One Has an inverse f-1 which is called the logarithmic function (loga) f-1(x) = y f(y) = x ay = x logax = y

  6. Example Find: log10(0.001) log216

  7. Log Graph Reflection of exponential function about the line y = x Domain (0, ∞) Range (-∞,∞)

  8. Laws of Logarithms loga(xy) = logax + logay loga(x/y) = logax – logay logaxr = rlogax

  9. Example Evaluate log280 – log25

  10. e y = ax Many formulas in calculus are greatly simplified if we use a base a such that the slope of the tangent line at y = 1 is exactly 1 For y = 2x, slope at y = 1 is .7 For y = 3x, slope at y = 1 is 1.1 Value of a lies between 2 and 3 and is denoted by the letter e e = 2.71828

  11. Example Graph y = ½ e-x – 1 and find the domain and range

  12. Natural log (ln) Log with a base of e logex= lnx lnx = y ey = x

  13. Properties of Natural Logs ln(ex) = x elnx = x ln e = 1

  14. Example Find x if lnx = 5

  15. Example • Solve e5 – 3x = 10

  16. Example • Express ln a + ½ ln b as a single logarithm

  17. Expression y = logax ay = x ln ay = ln x y ln a = ln x y = ln x/ ln a logax = ln x/ ln a if a ≠ 0

  18. Example • Evaluate log85

  19. Example • The half-life of a radioactive substance given by f(t) = 24 ∙ 2-t/25 Find the inverse

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