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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?.

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5.2 Logarithmic Functions & Their Graphs

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

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

  5. Evaluating Logs f(x) = log232 Step 1- rewrite it as an exponential equation. f(x) = log42 2y = 32 4y = 2 22y = 2 2y = 2 y = 1 Step 2- make the bases the same. 2y = 25 f(x) = log31 Therefore, y = 5 3y = 1 y = 0 f(x) = log10(1/100) 10y = 1/100 10y = 10-2 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?

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