1Summer MA 15200 Lesson 21 Section 4.2 and Applied Problems Remember the following information about inverse functions. 1. In order for a function to have an inverse, it must be one-to-one and pass a horizontal line test. 2. The inverse function can be found by interchanging x and y in the function’s equation and solving for y. 3. If 1() , then ()fab fba−==. The domain of f is the range of 1f− and the range of f is the domain of 1f−. 4. The compositions 11( ( )) and ( ( ))ffx ffx−− both equal x. 5. The graph of 1f− is the reflection of the graph of f about the line yx= . Because an exponential function is 1-1 and passes the horizontal line test, it has an inverse. This inverse is called a logarithmic function. I Logarithmic Functions According to point 2 above, we interchange the x and y and solve for y to find the equation of an inverse function. ()xfxb= exponential function inverse functionyxb= How do we solve for y? There is no way to do this. Therefore a new notation needs to be used to represent an inverse of an exponential function, the logarithmic function. Definition of Logarithmic Function For 0 and 0 ( 1)log is equivalent to ybxbbyxxb>>≠== The function ( ) logbfxx= is the logarithmic function with base b. The equation logbyx= is called the logarithmic form and the equation yxb= is called the exponential form. The value of y in either form is called a logarithm. Note: The logarithm is an exponent. Exponential Form Logarithmic Form ybx = xyblog= argument base exponent exponent base argument In this form, the y value representing the exponent is called a logarithm.2Ex 1: Convert each exponential form to logarithmic form and each logarithmic form to exponential form. 381log )532log )6418 )525 )16943 )813 )21222124=⎟⎠⎞⎜⎝⎛====⎟⎠⎞⎜⎝⎛=−fedcba 215log )5=g II Finding logarithms Remember: A logarithm is an exponent. Ex 2: Find each logarithm. 1032015) log 100,000) log 27) log 1) log 15abcd 121) log144e 723)4)(2))log(2 )12) log 200rpyqxhm xia pjmnkrs++=+===3 4123) log 64) log 32) log 81fgh III Basic Logarithmic Properties 1. log 1bb = Since the first power of any base equals that base, this is reasonable. 2. log 1 0b= Since any base to the zero power is 1, this is reasonable. The exponential function ( ) or xxfxb yb== and the logarithmic function 1() log or logbbfxxyx−== are inverses. 11( ( )) and ( ( ))ffx x ffx x−−== This leads to 2 more basic logarithmic properties. 3. logxbbx= This is a composition function where 1() and () logxbfxb fx x−==. 11(()) ( ) logxxbffx f b b x−−=== (the exponent) 4. logbxbx= This is a composition function where 1() and () logxbfxb fx x−==. log1( ( )) (log )bxbffx f xb x−===(the number or argument) Ex 3: Simplify using the basic properties of logarithms. 1243log 4510) log 1) log 3) 12) log 10abcd====4Ex 4: Simplify, if possible. =−=−)100log( )1log ))4(ba IV Graphs of Logarithmic Functions Below is a graph of yxxy 2 inverse,it and 2 == . If you imagine the line y = x, you can see the symmetry about that line. Below are both graphs on the same coordinate system along with y = x. Characteristics of a logarithmic Graph: The inverse of this function, ybx = , has a graph with the following characteristics. 1. The x-intercept is (1, 0). 2. The graph still is increasing if b > 1, decreasing if 0 < b < 1. 3. The domain is all positive numbers, so the graph is to the right of the y-axis. (The range is all real numbers.) 4. The y-axis is an asymptote. Remember that bases must be positive and the argument values (the numbers) must be positive. (1, 2) (2, 1)5V Common Logarithms A logarithmic function with base 10 is called the common logarithmic function. Such a function is usually written without the 10 as the base. 10log is equivalent to logxx A calculator with a key can approximate common logarithms. Put the number (argument) in the calculator, press the common log key. Ex 5: Find each common logarithm without a calculator. ===001.0log )1001log b)1000log )ca Ex 6: Use a calculator to approximate each common logarithm. Round to 4 decimal places. ) log0.025) log43.8ab Using the basic properties with base 10, we get the following properties. 1. log10 1= 2. log1 0= 3. log10xx= 4. log10xx= VI Natural Logarithms A logarithm function with base e is called the natural logarithmic function. Such a function is usually written using ln rather than log and no base shown. ln is equivalent to logexx A calculator with a key can approximate natural logarithms. Put the number (argument) in the calculator, press the natural log key. log ln6Ex 7: Use a calculator to approximate each natural logarithm. Round to 4 decimal places. ) ln0.988) ln2008ab Using the basic properties with base e, we get the following properties. 1. ln 1e = 2. ln1 0= 3. lnxex= 4. ln xex= VII Modeling with logarithmic functions The function ( ) 29 48.8log( 1)fx x=+ + gives the percentage of adult height attained by a boy who is x years old. Ex 8: Approximately what percentage of his adult height has a boy of age 11 acheived? (Notice: This model uses a common log.) Round to the nearest tenth of a percent. The function ( ) 13.4ln 11.6fx x=− models the temperature increase in degrees Fahrenheit after x minutes in an enclosed vehicle when the outside temperature is from 72° to 96°. Ex 9: Use the function above to approximate the temperature increase after 45 minutes. Round to the nearest tenth of a degree.7VIII Applied Problems For part of your homework on this lesson, you will have a worksheet of applied problems that use exponential or logarithmic formulas. You will need to print off the worksheet, the formula sheet, and/or the answer sheet from the course web page (under other information, worksheets). In order to solve the applied problems in this lesson, a student must know how to use the log, ln, xe , and power key functions on a scientific calculator. Ex 10: Half-Life of an Element The half-life of an element is the amount of time necessary for the element to decay to half the original amount. Uranium is an example of an element that has a half-life. The half-life of radium is approximately 1600 years. The formula used to find the amount of radioactive material present at time t, where A0 is
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