1 / 19

Exponential/ Logarithmic

This guide covers exponential and logarithmic functions, exploring their definitions, properties, and related laws. Exponential functions are characterized by their one-to-one nature and inverse relationships with logarithmic functions. The domain and range are identified, alongside different behaviors depending on the base (a). Key exponential laws are presented, along with examples illustrating logarithmic evaluations and the natural logarithm. Include practical applications such as solving equations and modeling real-world phenomena, like half-life in radioactive decay.

evania
Télécharger la présentation

Exponential/ Logarithmic

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related