Presentation on theme: "6.4 Logarithmic Functions"— Presentation transcript:
16.4 Logarithmic Functions MAT 204 F086.4 Logarithmic FunctionsIn this section, we will study the following topics:Evaluating logarithmic functions with base aGraphing logarithmic functions with base aEvaluating and graphing the natural logarithmic functionSolving logarithmic and exponential equations
2Logarithmic Functions MAT 204 F08Logarithmic FunctionsNow 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.
3Logarithmic Functions MAT 204 F08Logarithmic FunctionsFor 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.
4MAT 204 F08Subliminal MessageExponential and logarithmic functions of the same base are inverses.
5Logarithmic Functions (continued) Evaluate log28To evaluate log28 means to find the exponent such that 2 raised to that power gives you 8.
6Definition of a Logarithmic Function Logarithmic Functions (continued)The following definition demonstrates this connection between the exponential and the logarithmic function.Definition of a Logarithmic FunctionFor x > 0, a > 0, and a ≠ 1,y = logax if and only if x = ayWe read logax as “log base a of x”.
7y = logax if and only if x = ay Converting Between Exponential and Logarithmic Formsy = logax if and only if x = ayI. Write the logarithmic equation in exponential form.a)b)II. Write the exponential equation in logarithmic form.
8The plan is to convert to exponential form. Evaluating Logarithms w/o a CalculatorTo 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.
11The Common LogarithmThe 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.
12Graphs 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: Graph f(x) = 2x and g(x) = log2x in the same coordinate plane.To do this, we will make a table of values for f(x)=2x and then switch the x and y coordinates to make a table of values for g(x).
14Comparing the Graphs of Exponential and Log Functions Notice that the domain and range of the inverse functions are switched.The exponential function hasdomain (-, )range (0, )HORIZONTAL asymptote y = 0The logarithmic function hasdomain (0, )Range (-, )VERTICAL asymptote x = 0
15Transformations 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
16Transformations of Graphs of Logarithmic Functions
17The Natural Logarithmic Function In section 6.3, we saw the natural exponential function with base e. Its inverse is the natural logarithmic function with 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.
18Natural Log KeyTo evaluate the natural log using the TI-83/84, use the button.Notice, the 2nd function of this key is ________.
19Graph of the Natural Exponential and Natural Logarithmic Function f(x) = ex and g(x) = ln x are inverse functions and, as such, their graphs are reflections of one another in the line y = x.
20Evaluating the Natural Log Evaluate without using a calculator.a) b)c) d)e) f)
21Solving Logarithmic Equations Strategy for solving logarithmic equations: Change the equation from a log equation into an exponential equation, using one of the following forms:logax = y x = aylogx = y x = 10ylnx = y x = eyKeep in mind that the domain of the log function is x>0. Reject any extraneous solutions!!