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Logarithms Lesson PART I: Eponential Functions Eponential functions: These are functions where the variable is an eponent. The first type of eponential graph occurs when the base has a value greater than : y = - - - y = y = - - - - - - y = b b > When the base is greater than, the graph starts low and gets higher from left to right. - - - The second type occurs when the base has a value between 0 and. - - - - - - - - - y = b 0< b < When the base is between 0 and, the graph starts high and gets lower from left to right. - - - Notice that the point (0,) is common to all untransformed eponential graphs. This is because anything raised to the power of zero is one! You can use this feature as an anchor point when drawing these graphs. All eponential graphs have a horizontal asymptote. In the above graphs, the asymptote occurs along the -ais. (Equation: y = 0) The domain of untransformed eponential graphs is R since the graph goes left & right forever. The range is y >0. (The symbol is NOT used, due to the presence of the asymptote.) Remember all of the above rules are based on untransformed eponential graphs. Once transformations are involved, these points & lines will move. Pure Math 30: Eplained! www.puremath30.com 95
Logarithms Lesson PART II: Logarithmic Functions Logarithmic functions: A logarithmic function is the inverse of an eponential function. y = log b Variable To draw log graphs in your TI-3, you must type in log(variable) / log(base). Eample: To graph log, you would type in log() log() Base There are two basic types of log graphs you will need to memorize: The first type occurs when the base of the logarithm is bigger than. y = log y = log y = log - 3 5-3 5-3 5 y= log b > b When the base is greater than, the graph starts low and gets higher from left to right. - - - The second type is when the base of the logarithm is between 0 and. - y = log y = log y = log y = logb 0< b < 3 5-3 5-3 5 When the base is between 0 &, the graph starts high and gets lower from left to right. - - - Notice that the point (,0) is common to all untransformed log graphs. This occurs because a log graph the inverse of an eponential graph. So, if eponential graphs have the point (0,), it follows that log graphs should pass through (,0) log graphs have a vertical asymptote. In the above graphs, the asymptote occurs along the y-ais. (Equation: = 0) The domain of untransformed log graphs is > 0 since the graph is always to the right of the vertical asymptote. The range is y R since the graph goes up & down forever. If you ever have negative numbers, 0, or as a base, no graph eists since the logarithm is undefined. Remember that transformations will change the above values. is Pure Math 30: Eplained! www.puremath30.com 9
Logarithms Lesson I PART II: Logarithmic Functions Eample : Given a) Draw the graph Graph in your calculator as ( 5 ) ^ Use the window settings: : [-,, ] y: [-,, ] b) What is the domain & range The domain is ε R The range is y > 0 y= 5, answer the following: c) What is the equation of the asymptote? The asymptote is the -ais, so the equation is y = 0 - - - d) What are the & y intercepts? There is no -intercept due to the asymptote.. nd Find the y-intercept by using Trace Value = 0 in your TI-3. Answer = (0, ) e) What is the value of the graph when =? You could plug = into the equation and solve, but an easier way is to use the TI-3. Go nd Trace Value =. This will give you the resulting y-value automatically. Answer =.5 Eample : Given y =log a) Draw the graph Graph in your calculator as log() log() Use the window settings: : [0,, ] y: [-,, ] b) What is the domain & range The domain is > 0 due to the vertical asymptote at the y-ais. The range is y ε R c) What is the equation of the asymptote? The asymptote is the y-ais, so = 0 d) What are the & y intercepts? The intercept can be found by going nd Trace Zero in your TI-3. Answer = (, 0) There is no y-intercept due to the vertical asymptote at the y-ais. e) What is the value of the graph when =? Go nd Trace Value = Answer = 0.5, answer the following: 3 - - -3 - - - You will always have to type logarithms into your calculator as a fraction with one eception: a logarithm without a base, such as y = log, can be typed in as is and you ll get the proper graph. logarithms without bases are called common logarithms. they actually have a base of 0, It s just not written in. The log button on your calculator is a common logarithm. Pure Math 30: Eplained! www.puremath30.com 97
Function Graph Domain Range Equation of Asymptote -intercept y-intercept y-value when = y = 3 - - - - - - 3 - - - - - - y = 3 log. - - - 3 - - - - - - ( ) y = log 3 y = + ( ) y = log + Pure Math 30: Eplained! www.puremath30.com 9
Function Graph Domain Range y = 3 - - - 3 y. - - - - - - - = log 3 3 Notice how the base is greater than. - - - - - - - - - - - Equation of Asymptote -intercept y-intercept y-value when = ε R y > 0 y = 0 None 9 ε R y > 0 y = 0 None > 0 y ε R = 0 None 3.0 ε R y > 0 y = 0 None.5 9 ( ) y = log 3 > 3 y ε R = 3 None Undefined y = + ε R y > y = None 3 ( ) y = log + > - y ε R = - - 0.30 0.0 ε R y > - y = - - -3-3.75 Pure Math 30: Eplained! www.puremath30.com 99
Logarithms Lesson I PART iii: Eponential Regression Eponential Regression: Using the TI-3, it is possible to find an eponential equation from a list of data. Eample : Given the following data, determine the eponential regression equation. Step : Type: Stat Edit to bring up the list function of your calculator. Step : Fill in the -values for L and the y-values for L. (If there is data in the first list, put the cursor at the very top over L and type clear enter to empty the entire column.) y 0.3. 3.9.7 5 3.3 3.7 Step 3: Type nd Quit to return to the main screen. Step : Type Stat Calc EpReg (Just hit the zero button to bring it up quickly) Enter The screen that comes up net tells you the equation is of the form y = a(b), and the numerical values for a & b are given below. The eponential regression equation is y =.905(.077) Step 5: If you draw the graph, manually type this equation in as you would any other graph. *Alternatively, you could copy the equation by doing the following: Type: Y = Vars Statistics EQ RegEQ -0-5 5 0 - - Pure Math 30: Eplained! www.puremath30.com 00
Logarithms Lesson I PART iii: Eponential Regression Eample : Given the following table: a) Determine the eponential regression equation. Using the steps from the previous eample, the regression equation is: y = 3.00(.33) b) Draw the graph. y 5 5 5 7 7 0 03 3 7 30 00 300 00 00-0 -5 5 0 5 0 c) Find y when = 9 Now that you have the graph in your TI-3, type: nd Trace Value = 9 Answer: y = 99.79 d) Find when y = 5 In order to solve this, draw in a second graph, the line y = 5. 00 300 00 00 Now find the point of intersection of these two graphs, and the -value will be the solution. Answer: = 5.9-0 -5 5 0 5 0 Pure Math 30: Eplained! www.puremath30.com 0
Logarithms Lesson I PART iii: Eponential Regression Questions: Determine the eponential regression equation for each of the following sets of data. Graph each result... y a) Determine the regression y a) Determine the regression 0 5 equation. 0 93 equation. 0 3 990 9 0 3 9 39 9 b) Draw the graph. 3 b) Draw the graph. 5 3 0 5 3 3500 00 0 0 0 0 3000 500 000 500 000 500 3 5 0 c) Find y when = 9 (Nearest Hundredth) c) Find y when = 3 (Nearest Hundredth) d) Find when y = 33 (Nearest Hundredth) d) Find when y = 0000 (Nearest Hundredth) y 0 0.05 0. 0. 3 0.. 5 3. 3. a) Determine the regression y equation. 0 30.9 5.7 0 7.5 b) Draw the graph. 5. 0.3 5 5 3. a) Determine the regression equation. b) Draw the graph. 30 5 0 5 0 5 3 5 5 0 5 0 5 c) Find y when = 0 (Nearest Hundredth) c) Find y when = 3 (Nearest Hundredth) d) Find when y = 0.0 (Nearest Hundredth) d) Find when y = 7 (Nearest Hundredth) Pure Math 30: Eplained! www.puremath30.com 0
Logarithms Lesson I PART iii: Eponential Regression Answers:.. a) 5.5(.55) a) 77.97(.073) b) b) 3500 0 00 0 0 0 0 3000 500 000 500 000 500 c) Find y when = 9 y = 0.30 d) Find when y = 33. (By graphing) 3 5 0 c) Find y when = 3.9 d) Find when y = 0000 30.37 3.. a) 0.053(.333) a) 9.99(0.753) b) 5 b) 30 3 5 0 5 0 5 3 5 c) Find y when = 0.9 d) Find when y = 0.0 0.7 (By graphing) c) Find y when = 3 0.05 d) Find when y = 7 0.9 (By graphing) 5 0 5 0 5 Pure Math 30: Eplained! www.puremath30.com 03