1 / 18

5.2 Logarithmic Functions & Their Graphs

5.2 Logarithmic Functions & Their Graphs. Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and graph natural logs Use logarithmic functions to model and solve real-life problems. f(x) = 3 x. Is this function one to one?.

fathia
Download Presentation

5.2 Logarithmic Functions & Their Graphs

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. 5.2 Logarithmic Functions & Their Graphs Goals— Recognize and evaluate logarithmic functions with base a Graph Logarithmic functions Recognize, evaluate, and graph natural logs Use logarithmic functions to model and solve real-life problems.

  2. f(x) = 3x Is this function one to one? Horizontal Line test? Does it have an inverse?

  3. Logarithmic function with base “a” Definition For x > 0, a > 0, and a  1, y = logax if and only if x = ay The function given by f(x) = logax read as “log base a of x” is called the logarithmic function with base a.

  4. Writing the logarithmic equation in exponential form log168 = 3/4 log381 = 4 34 = 81 163/4 = 8 Writing an exponentialequation in logarithmic form 82 = 64 4-3 = 1/64 log 8 64 = 2 log4 (1/64) = -3

  5. Evaluating Logs Step 1: Rewrite the log problem as an exponential. 4y = 2 y = 5 (22)y = 21 y= log232 22y = 21 2y = 32 y= log42 y = 1/2 y = 1/2 Step 2: Rewrite both sides of the = with the same base. 3y = 1 y= log31 y = 0 y = 0 2y = 25 10y = 1/100 Therefore, y = 5 y= log101/100 10y = 10-2 y = -2 y = -2

  6. Evaluating Logs on a Calculator You can only use a calculator when the base is 10 f(x) = log x when x = 10 f(x) = 1 when x = 1/3 f(x) = -.4771 when x = 2.5 f(x) = .3979 when x = -2 f(x) = ERROR!!! Why???

  7. Properties of Logarithms • loga1 = 0 because a0 = 1 • logaa = 1 because a1 = a • logaax = x and alogax = x • logax = logay, then x = y

  8. Simplify using the properties of logs Rewrite as an exponent 4y = 1 So y = 0 log41 Rewrite as an exponent 7y = 7 So y = 1 • log77 • 6log620

  9. Use the 1-1 property to solve • log3x = log312 • x = 12 • log3(2x + 1) = log3x • 2x + 1 = x • x = -1 • x2 - 6 = 10 • x2 = 16 • x = 4 • log4(x2 - 6) = log4 10

  10. f(x) = 3x Graphs of Logarithmic Functions So, the inverse would be g(x) = log3x Make a T chart Domain— Range? Asymptotes?

  11. Graphs of Logarithmic Functions g(x) = log4(x – 3) Make a T chart Domain— Range? Asymptotes?

  12. Graphs of Logarithmic Functions g(x) = log5(x – 1) + 4 Make a T chart Domain— Range? Asymptotes?

  13. Natural Logarithmic Functions • The function defined by f(x) = loge x = ln x, x > 0 is called the natural logarithmic function.

  14. Evaluate f(x) = ln x when x = 2 f(x) = .6931 when x = -1 f(x) = Error!!! Why???

  15. Properties of Natural Logarithms ln 1 = 0 because e0 = 1 ln e = 1 because e1 = e ln ex = x and elnx = x (Think…they are inverses of each other.) If ln x = ln y, then x = y

  16. Use properties of Natural Logs to simplify each expression ln (1/e) = ln e-1 = -1 eln 5 = 5 2 ln e = 2

  17. Graphs of Natural Logs g(x) = ln(x + 2) Make a T chart 2 Undefined 3 4 Domain— Range? Asymptotes?

  18. Graphs of Natural Logs g(x) = ln(2 - x) Make a T chart 2 Undefined 1 0 Domain— Range? Asymptotes?

More Related