Presentation on theme: "Logarithmic Functions"— Presentation transcript:
1 Logarithmic Functions Section 4.2 Part 1Logarithmic FunctionsObjectives:Change from logarithmic to exponential form.Change from exponential to logarithmic form.Evaluate logarithms.Use basic logarithmic properties.
2 Definition of a Logarithmic Function For x > 0 and b > 0, b = 1,y = logb x is equivalent to by = x.The function f (x) = logb x is thelogarithmic function with base b.
3 Location of Base and Exponent in Exponential and Logarithmic Forms Logarithmic form: y = logb x Exponential Form: by = x.ExponentExponentBaseBase
4 Practice #1 Write each equation in its equivalent exponential form. a. 2 = log5 x b. 3 = logb 64 c. log3 7 = y
5 With the fact that y = logb x means by = x, SolutionWith the fact that y = logb x means by = x,a. 2 = log5 x means 52 = x.Logarithms are exponents.b. 3 = logb 64 means b3 = 64.Logarithms are exponents.c. log3 7 = y or y = log3 7 means 3y = 7.
6 Practice #2Evaluatea. log2 16 b. log c. log25 5Solution
7 First, rewrite each expression as a logarithmic equation and convert SolutionFirst, rewrite each expression as a logarithmic equation and convertto an exponential equation.Logarithmic EquationExponential EquationQuestion Needed for EvaluationEvaluationa. log216 = x2x = 162 to what power is 16?4b. log39 = x3x = 93 to what power is 9?2c. log255 = x25x = 525 to what power is 5?Since you must take the square root of 25 to get 5, that is the same as an exponent of ½. The answer is ½ .
8 Basic Logarithmic Properties Involving One logb b = 1 because 1 is the exponent to which b must be raised to obtain b. (b1 = b).logb 1 = 0 because 0 is the exponent to which b must be raised to obtain 1. (b0 = 1).
9 Inverse Properties of Logarithms For x > 0 and b 1,logb bx = x The logarithm with base b of b raised to a power equals that power.b logb x = x b raised to the logarithm with base b of a number equals that number.