Topic 5: Logarithmic Regression Logarithmic Functions

I can graph data, and determine the logarithmic function that best approximates the data. I can interpret the graph of a logarithmic function that models a situation, and explain the reasoning. I can solve, using technology, a contextual problem that involves data that is best represented by graphs of logarithmic functions, and explain the reasoning.

Information A scatter plot is a set of points on a grid, used to visualize a possible trend in the data. A logarithmic regression function is the equation of a curve of best fit when the scatter plot looks like a logarithmic function. Technology uses logarithmic regression to determine the equation that balances the points in the scatter plot on both sides of the curve. The general form of a logarithmic function is . The calculator, however, generates the regression equation using natural logs, resulting in an equation of the following form:

Example 1 Using logarithmic regression to solve a problem Lydia is researching the rise in tuition fees for post- secondary education in Alberta. She found some data that used the tuition fees in 1992 as the benchmark, assigning them a value of 100%. The tuition fees in all other years are compared with the tuition fees in 1992. a) Construct a scatter plot where year is a function of tuition cost. Label the scatter plot, including your window settings. x (L1) y (L2) Since the question says ‘year as a function of tuition,’ tuition is the independent variable (x) and year is the dependent variable (y).

Example 1 Using logarithmic regression to solve a problem a) Construct a scatter plot where year is a function of tuition cost. Label the scatter plot, including your window settings. Remember the steps: 1. Enter data into L1 and L2 by pressing STAT and 1:Edit. 2. Adjust the window settings so that all of the data values will be in view. 3. Press GRAPH to display your scatterplot. (if it doesn’t show up, make sure Plot 1 is turned on under 2nd Y=).

Example 1 Using logarithmic regression to solve a problem b) Determine the equation of the logarithmic regression function that models her data. Sketch the graph. Remember the steps: 1. Press STAT and  to get to the calculate menu. Select 9: LnReg. 3. Press ENTER to see your Equation. Then press Graph to see the graph. 2. Press VARS and then  to get to Y-Vars. Select 1:Function and then 1:Y1. (This calculates the regression equation and then puts the equation in Y1.) y = 1937.42+11.78lnx

Example 1 Using logarithmic regression to solve a problem c) Is the regression equation a good fit for the data? y = 1937.42+11.78lnx The graph of the equation matches very closely to the scatterplot, so yes. Also, if you were to look at the correlation coefficient, it is very close to 1. This also would indicate a very good fit.

Example 2 Using logarithmic regression to solve a problem Caffeine is found in coffee, tea, and soft drinks. Many people find that caffeine makes it difficult for them to sleep. The following data was collected in a study to determine how quickly the human body metabolizes caffeine. Each person started with 200 mg of caffeine in her or his bloodstream, and the caffeine level was measured at various times. a) Determine the equation of the logarithmic regression function for the data representing time as a function of caffeine level. x (L1) y (L2) Since the question says ‘time as a function of caffeine level,’ caffeine is the independent variable (x) and time is the dependent variable (y).

Example 2 a) Determine the equation of the logarithmic regression function for the data representing time as a function of caffeine level. Enter data (STAT, 1:Edit). 4. Calculate the regression equation. 2. Adjust window. y = 43.79 – 8.21lnx 3. Graph.

Example 2 b) Determine the time it takes for an average person to metabolize 50% of the caffeine in her or his bloodstream. Round your answer to the nearest tenth of an hour. 50% of the original 200mg of caffeine is 100mg. The question is asking what the y-value is for an x-value of 100. When given an x-value, calculate the y-value by pressing 2nd Trace and selecting 1:Value. y = 43.79 – 8.21lnx It takes 6.0 hours for the average person to metabolize 50% of the caffeine. Enter the x-value of 100 and press ENTER.

Example 2 c) Paula drank a cup of coffee what contained 200mg of caffeine at 10:00 am. How much caffeine will be in her bloodstream 11 hours later, at 9:00 pm that evening? Round your answer to the nearest milligram. The question is asking what the x-value is for an y-value of 11. y = 43.79 – 8.21lnx When given a y-value, enter it into Y2. Then press GRAPH. To find the intersection point (the point were the graph is equal to 11), press 2nd TRACE 5: Intersect. You will have to press Enter 3 times. At 10pm , there is 54 mg of caffeine left in her bloodstream.

Example 2 d) Paula is only able to sleep if her caffeine levels are below 70 mg. How long will it take for her body to metabolize the caffeine in her bloodstream so that she can sleep, to the nearest hour? 70 mg is an x-value. We need to find the y-value. Enter 2nd TRACE 1:Value. Enter an x-value of 70. y = 43.79 – 8.21lnx This takes 9 hours for the caffeine level to be low enough for sleep.

Example 2 e) Using your regression equation, determine the x-intercept to the nearest tenth. Explain what the x-intercept represents in the given context. To find the x-intercept, press 2nd TRACE and select number 2: zero. Move the cursor close to the x-intercept. You must move it to the left of the intercept and press enter. Then you must move it to the right of the intercept and press enter. Then move the cursor close to the x-intercept and press enter one final time. y = 43.79 – 8.21lnx The x-intercept is 207.7mg. It actually represents the amount of caffeine at time 0 (based on my line of best fit).

Need to Know A logarithmic function may be a good model for a set of data if the points on a scatter plot form an increasing or decreasing curve, where the domain is restricted to the set of positive real numbers. A logarithmic regression function is the equation of a curve of best fit when the scatter plot looks like a logarithmic function. The general form of a logarithmic function is . However, the calculator generates the regression equation using natural logs, giving an equation of the following form:

You’re ready! Try the homework from this section. Need to Know A logarithmic curve of best fit can be used to predict values that are not recorded or plotted (interpolation or extrapolation). Predictions can be made by reading values from the curve of best fit on a scatter plot or by using the equation of the logarithmic regression function.