Presentation on theme: "3.3 Logarithmic Functions and Their Graphs"— Presentation transcript:
13.3 Logarithmic Functions and Their Graphs We learned that, if a function passes the horizontal line test, then the inverse of the function is also a function.So, an exponential function f(x) = bx, has an inverse that is a function.This inverse is the logarithmic function with base b, denoted or
2Changing Between Logarithmic and Exponential Form If x > 0 and 0 < b ≠ 1, thenif and only ifThis statement says that a logarithm is an exponent. Because logarithms are exponents, we can evaluate simple logarithmic expressions using our understanding of exponents.
3Evaluating Logarithms a.) because . b.) because . c.) because d.) because e.) because
4Basic Properties of Logarithms For 0 < b ≠ 1 , x > 0, and any real number y,logb1 = 0 because b0 = 1.logbb = 1 because b1 = b.logbby = y because by = by.because
5Evaluating Logarithmic and Exponential Expressions (a) (b) (c)
6Common Logarithms – Base 10 Logarithms with base 10 are called common logarithms.Often drop the subscript of 10 for the base when using common logarithms.The common logarithmic function:y = log x if and only if 10y = x.
7Basic Properties of Common Logarithms Let x and y be real numbers with x > 0.log 1 = 0 because 100 = 1.log 10 = 1 because 101 = 10.log 10y = y because 10y = 10y .because log x = log x.Using the definition of common logarithm or these basic properties, we can evaluate expressions involving a base of 10.
8Evaluating Logarithmic and Exponential Expressions – Base 10 (a) (b) (c) (d)
9Evaluating Common Logarithms with a Calculator See example 4 p. 303
10Solving Simple Logarithmic Equations Solve each equation by changing it to exponential form:a.) log x = 3 b.)a.) Changing to exponential form, x = 10³ = 1000.b.) Changing to exponential form, x = 25 = 32.
11Natural Logarithms – Base e Because of their special calculus properties, logarithms with the natural base e are used in many situations.Logarithms with base e are natural logarithms.We use the abbreviation “ln” (without a subscript) to denote a natural logarithm.
12Basic Properties of Natural Logarithms Let x and y be real numbers with x > 0.ln 1 = 0 because e0 = 1.ln e = 1 because e1 = e.ln ey = y because ey = ey.eln x = x because ln x = ln x.
13Evaluating Logarithmic and Exponential Expressions – Base e (a) (b) (c)
14Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function.a.) g(x) = ln (x + 2)The graph is obtained by translating the graph of y = ln (x) two units to the LEFT.
15Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function.b.) h(x) = ln (3 - x)The graph is obtained by applying, in order, a reflection across the y-axis followed by a transformation three units to the RIGHT.
16Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function.c.) g(x) = 3 log xThe graph is obtained by vertically stretching the graph of f(x) = log x by a factor of 3.
17Transforming Logarithmic Graphs Describe how to transform the graph of y = ln x or y = log x into the graph of the given function.d.) h(x) = 1+ log xThe graph is obtained by a translation 1 unit up.
18Measuring Sound Using Decibels The level of sound intensity in decibels (dB) iswhere (beta) is the number of decibels, I is the sound intensity in W/m², and I0 = 10 – 12 W/m² is the threshold of human hearing (the quietest audible sound intensity).
19More Practice!!!!!Homework – Textbook p. 308 #2 – 52 even.