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Logarithmic Functions & Their Graphs

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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Logarithmic Functions & Their Graphs

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  1. 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 “b” The logarithm to the base “b” of a positive number y is defined as follows: If y = bx , then logby = x 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 Write the exponential equation in logarithmic form 82 = 64 4-3 = 1/64

  5. Evaluating Logs f(x) = log232 Step 1- rewrite it as an exponential equation. f(x) = log42 Step 2- make the bases the same. f(x) = log31 f(x) = log10(1/100)

  6. Common Logarithms • Use base 10 • Written – • Log10y is the same as log y • Only Common Logs can be evaluated using a calculator.

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

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

  9. 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

  10. 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

  11. f(x) = 3x Graphs of Logarithmic Functions Now graph g(x) = log3x Make a T chart Domain— Range? Asymptotes?

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

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

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

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

  16. 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

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

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

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

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