1 / 29

Precalculus

Precalculus. Exponential & Logarithmic Functions. Exponential Functions. The exponential function f with base a is defined as f(x) = a x . where a > 0, a ≠ 1, and x is any real number. For a > 1, the graph of y = a x is ___________ over its domain.

tyson
Télécharger la présentation

Precalculus

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. Precalculus Exponential & Logarithmic Functions

  2. Exponential Functions • The exponential function f with base a is defined as f(x) = ax . where a > 0, a ≠ 1, and x is any real number. • For a > 1, the graph of y = ax is ___________ over its domain. • For a > 1, the graph of y = a-x is __________ over its domain. • For the graph of y = ax or y = a-x, a > 1, the domain is _______, the range is ________, and the intercept is __________. Also, both graphs have __________ as a horizontal asymptote.

  3. Natural Exponential Functions • The natural exponential function f has base e and is defined as f(x) = ex . • The graph of y = ex is ___________ over its domain. • The graph of y = e-x is __________ over its domain. • For the graph of y = ex or y = e-x, the domain is _______, the range is ________, and the intercept is __________. Also, both graphs have __________ as a horizontal asymptote.

  4. Compounding Interest • After t years, the balance A in an account with principal P and annual interest rate r (expressed as a decimal) … • For n compoundings per year: • For continuous compounding: • Note: Continuous Compounding yields more than quarterly or monthly compounding …

  5. Examples • f(-x) = Reflect about the y-axis … • -f(x) = Reflect about the x-axis … • Given y = ax … • __________ is a reflection about the y-axis. • __________ is a reflection about the x-axis.

  6. Examples • Graph the following functions …

  7. Examples • Determine if the following functions are the same …

  8. Examples • Radium 226 has a half life of 1620 years. Its mass (Q) is given by the following equation (t = years) … • Find the initial quantity. • Find the quantity after 1000 yrs. • Graph the function over t = 0 to t = 5000. • When will the quantity be 0 grams?

  9. Logarithmic Functions • The logarithmic function with base a is defined as y = logax for x > 0, a > 0, a ≠ 1, if and only if x = ay. • The equation x = ay in exponential form is equivalent to the equation y = logax in logarithmic form. • The logarithmic function with base a is the inverse of the exponential function f(x) = ax. • Example: Use the definition of a logarithmic function to evaluate log5125 …

  10. Properties of Logarithms • loga1 = ____________ (because a0 = 1) • logaa = ____________ (because a1 = a) • logaax = ____________ (Inverse Properties) • alogax = ____________ (Inverse Properties • If logax = logay , then _____________ (1-1 Property)

  11. Graphs of Logarithms • To sketch y = logax , use the fact that the graphs of inverse functions are reflections of each other in the line y = x. • For the graph of y = logax , a > 1, the domain is _________, the range is __________, and the intercept is __________. • Also, the graph has __________ as a vertical asymptote.

  12. Examples • Graph the following functions …

  13. Examples • Graph the following functions …

  14. Properties of Natural Logarithms • ln1 = ____________ (because e0 = 1) • lne = ____________ (because e1 = e) • lnex = ____________ (Inverse Properties) • elnx = ____________ (Inverse Properties • If lnx = lny , then _____________ (1-1 Property)

  15. Examples • Write the logarithmic equation in exponential form … • Write the exponential equation in logarithmic form …

  16. Examples • Solve the equation for x …

  17. Examples • Find the domain, vertical asymptote, and x-intercept of the following functions …

  18. Properties of Logarithms • Common Logarithms = base 10 • Natural Logarithms = base e

  19. Change of Base Formula • Let a, b, and x be positive real numbers such that a ≠ 1 and b ≠ 1. Then logax can be converted to a different base using any of the following formulas.

  20. Examples • Evaluate the following logarithms using the change-of-base formula …

  21. Examples • Rewrite the logarithm as a multiple of a common logarithm and a natural logarithm …

  22. Properties of Logarithms • Let a be a positive number such that a ≠ 1, and let n be a real #. If u and v are positive real #’s, the following are true …

  23. Examples • Use the properties of logarithms to write the expression as a sum, difference, and/or constant multiple of a logarithm …

  24. Examples • Write the expression as the logarithm of a single quantity …

  25. Examples • Find the exact value of the logarithm without using a calculator …

  26. Solving Exponential & Logarithmic Equations • One-to-One Properties … ax = ay if and only if x = y logax = logay if and only if x = y • Inverse Properties …

  27. Solving Exponential & Logarithmic Equations • Rewrite the equation in a form to use the one-to-one properties of exponential or logarithmic functions. • Rewrite an exponential equation in logarithmic form and apply the inverse property of logarithmic functions. • Rewrite a logarithmic equation in exponential form and apply the inverse property of exponential functions.

  28. Examples • Solve the following exponential equations …

  29. Exponential & Logarithmic Models • Exponential Growth • Exponential Decay • Gaussian Model • Logistic Growth • Logarithmic Model

More Related