Intermediate Algebra Section 9.3 Logarithmic Functions We have studied inverse functions, learning when they eist and how to find them. If we look at the graph of the eponential function, f ( ) = a, where a > 0, a 1(shown below for a = ), we see that the graph of this function passes the horizontal line test. In fact, the graph of the eponential function for any a > 0 will pass the horizontal line test, so we know that there is an inverse function for f ( ) = a, where a > 0, a 1. f ( ) = a, where a = To find the inverse, recall that we interchange and y and solve for y. If we do that here, we get f ( ) = y = = We cannot solve this for y. We can see that y is the eponent that is raised to to obtain. We use the term logarithm to denote y. We epress this as y = log, and this means that y is the eponent that must be raised to to obtain. So, a logarithm is merely an eponent. y
Section 9.3 Logarithmic Functions page Logarithmic Definition y For a > 0 and a 1, then y = log a means = a for every > 0 and every real number y. As you work, try using this sentence: The power to which I raise a to to get is y. Eample: Write the following as an eponential epression. 1 a) log 3 = 5 b) log5 5 = c) log10 10 = 1 d) log y = e 7
Section 9.3 Logarithmic Functions page 3 Eample: a) Write the following as a logarithmic epression. 3 5 = 15 b) 10 1 = 100 c) 3 = d) 81 4 1 5 e = y Eample: Find the value of the following logarithms. a) log3 9 b) log7 9 c) log 66 d) log 41
Section 9.3 Logarithmic Functions page 4 To sketch the graph of the logarithmic function, you can translate into the eponential form and plot points and use characteristics of the logarithmic graph, or you can graph the inverse function and reflect about the line y =. For eample, the graph of f ( ) = is shown. We will graph its 1 inverse function f ( ) = log and look at the characteristics of the logarithmic graph. f ( ) = Domain: (, ) Range: ( ) Intercept: ( ) Increasing 1 f ( ) = log Domain: ( 0, ) 0, Range: (, ) 0,1 Intercept: ( 1,0 ) Increasing Eample: Find the domain of the following logarithmic functions. a) f ( ) = log ( ) b) H ( ) = log ( 5) 3 4
Section 9.3 Logarithmic Functions page 5 Eample: Graph. g( ) = log 4 y 18 16 14 1 10 8 6 4-18 -16-14 -1-10 -8-6 -4-4 6 8 10 1 14 16 18 - -4-6 -8-10 -1-14 -16-18
Section 9.3 Logarithmic Functions page 6 g = log Eample: Graph. ( ) 1 y 18 16 14 1 10 8 6 4-18 -16-14 -1-10 -8-6 -4-4 6 8 10 1 14 16 18 - -4-6 -8-10 -1-14 -16-18
Section 9.3 Logarithmic Functions page 7 Common Logarithms Logarithms with base 10 are called common logarithms and are f = log = log. written without the eplicit base. That is, ( ) 10 If the base of the logarithm is the irrational number e, the logarithm is called a natural logarithm and is written f ( ) = log = ln. We refer to this logarithm as the natural logarithmic function because it arises in a lot of natural events. Natural Logarithms ln means log e A calculator can be used to evaluate common logarithms and natural logarithms. Eample: Use a calculator to evaluate the following logarithmic epressions. Round answers to three decimal places. a) log106 b) log 0.78 c) ln 0.78 e Eample: Solve the following logarithmic equations. log4 8 + 10 = 3 b) log381 = a) ( )
Section 9.3 Logarithmic Functions page 8 c) ln = 10 An earthquake whose seismographic reading measures millimeters has magnitude M given by ( ) log M = 10 3 3 where 10 is the reading of a zero-level earthquake 100 kilometers from its epicenter. Eample: According to the United States Geological Survey, an earthquake on January 31, 1906 off the coast of Ecuador had a magnitude of 8.8. What was the seismographic reading 100 kilometers from its epicenter?