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6.4 Logarithmic Functions

6.4 Logarithmic Functions. In this section, we will study the following topics: Evaluating logarithmic functions with base a Graphing logarithmic functions with base a Evaluating and graphing the natural logarithmic function Solving logarithmic and exponential equations.

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6.4 Logarithmic Functions

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  1. 6.4 Logarithmic Functions In this section, we will study the following topics: Evaluating logarithmic functions with base a Graphing logarithmic functions with base a Evaluating and graphing the natural logarithmic function Solving logarithmic and exponential equations

  2. Logarithmic Functions Now that you have studied the exponential function, it is time to take a look at its INVERSE: THE LOGARITHMIC FUNCTION. In the exponential function, the independent variable (x) was the exponent. So we substituted values into the exponent and evaluated it for a given base. Exponential Function: f(x) = 2x, f(3) = 23= 8.

  3. Logarithmic Functions For the inverse function (LOGARITHMIC FUNCTION), the base is given and the answer is given, so to evaluate a logarithmic function is to find the exponent. That is why I think of the logarithmic function as the “Guess That Exponent”function. Warm Up: Give the value of ? in each of the following equations.

  4. Subliminal Message Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses. Exponential and logarithmic functions of the same base are inverses.

  5. Logarithmic Functions(continued) Evaluate log28 To evaluate log28 means to find the exponent such that 2 raised to that power gives you 8.

  6. Logarithmic Functions(continued) The following definition demonstrates this connection between the exponential and the logarithmic function. We read logax as “log base a of x”.

  7. Converting Between Exponential and Logarithmic Forms y = logax if and only if x = ay I. Write the logarithmic equation in exponential form. a) b) II. Write the exponential equation in logarithmic form. a) b)

  8. Evaluating Logarithms w/o a Calculator To evaluate logarithmic expressions by hand, we can use the related exponential expression. Example: Evaluate the following logarithms: The plan is to convert to exponential form.

  9. Evaluating Logarithms w/o a Calculator(cont.)

  10. Evaluating Logarithms w/o a Calculator Okay, try these.  e) f) g) h)

  11. The Common Logarithm The common logarithm has a base of 10. If the base of a logarithm is not indicated, then it is assumed that the base is 10.

  12. Graphs of Logarithmic Functions • Since the logarithmic function is the _______________ of the exponential function (with the same base), we can use what we know about inverse functions to graph it. • Example:Graphf(x) = 2xand g(x) = log2xin the same coordinate plane. • To do this, we will make a table of values for f(x)=2x and then switchthe x and y coordinates to make a table of values for g(x).

  13. f(x) = 2x y =x g(x)= log2x Graphs of Logarithmic Functions(continued) Inverse functions f(x) = 2x g(x) = log2x

  14. Comparing the Graphs of Exponential and Log Functions • Notice that the domain and range of the inverse functions are switched. • The exponential function has • domain (-, ) • range (0, ) • HORIZONTAL asymptote y = 0 • The logarithmic function has • domain (0, ) • Range (-, ) • VERTICAL asymptote x = 0

  15. Transformations of Graphs of Logarithmic Functions The same transformations we studied earlier also apply to logarithmic functions. Look at the following shifts and reflections of the graph of f(x) = log2x. The new vertical asymptote is x = -2

  16. Transformations of Graphs of Logarithmic Functions

  17. The Natural Logarithmic Function In section 6.3, we saw the natural exponential function with base e. Its inverse is the natural logarithmic functionwith base e. Instead of writing the natural log as logex, we use the notation , which is read as “the natural log of x” and is understood to have base e.

  18. Natural Log Key • To evaluate the natural log using the TI-83/84, use the  button. • Notice, the 2nd function of this key is ________.

  19. Graph of the Natural Exponential and Natural Logarithmic Function f(x) = ex and g(x) = ln x areinverse functions and, as such, their graphs are reflections of one another in the line y = x.

  20. Evaluating the Natural Log Evaluate without using a calculator. a) b) c) d) e) f)

  21. Solving Logarithmic Equations Keep in mind that the domain of the log function is x>0. Reject any extraneous solutions!!

  22. Examples of solving log equations

  23. More examples of solving log equations

  24. End of Section 6.4

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