Course Information[br][br][list][*][b]Course:[/b] Mathematics[br][/*][*][b]Class:[/b] 11[br][/*][*][b]Duration:[/b] 40+40 min[/*][*][b]Technological Equipment:[/b][i] [/i]An overhead projector and students'smart phones[/*][/list][br][b]Course Content[br][/b][br]Logarithmic functions that are the inverse of exponential functions[br][br][b]Learning Outcomes[br][/b][br]Students will be able to:[br][list] [*]observe the changes, increases and decreases in function  [math]f\left(x\right)=log_ax[/math]for 0<a<1 and a>1,[/*][*]draw the graph of [math]f\left(x\right)=log_a\left(x+b\right)+c[/math][/*][*]compare and contrast logarithmic functions and exponential functions and state on which line they are symmetrical to each other. [/*][/list][b]Course Objectives and Assessment[/b][br][br]Students will: [list][*]discover the concept of logarithmic functions, [/*][*]draw a standard graph of a logarithmic function, [/*][*]find out the right equation of the line on which exponential functions and logarithmic functions which have the same base are symmetrical.[/*][/list][br][b]Learning Strategies[/b][br][list] [*]The online version of Geogebra will be used and students will be first reminded of what they already know about the drawing process of an exponential function. [/*][*]There will be a transition from the definition of inverse functions to logarithmic functions.[/*][*]The online version of Geogebra will be used; a logarithmic function will be drawn, and students will be asked to draw the logarithmic functions they are given with the help of the drawing they have just examined. [/*][*]Finally, students will compare and contrast exponential and logarithmic functions. [/*][*]A “question and answer” method will be frequently used in class. At the end of the session, students will be asked to draw their graphs at their own learning pace on their own tablet PCs. [/*][/list][br][b]Resources [/b][br]The online version of Geogebra program will be used. [br][br][b]Integration of Technology[/b] [list][*]Since I have already implemented Geogebra in classroom before, I have already taken the necessary precautions against possible obstacles that might occur (e.g. in case the screen-touch freezes, a mouse and a keyboard will be used to continue the class).[/*][*]We have a problem-free internet access in our smart board.[/*][*]The biggest obstacle that we could ever face is a power cut, but in such a case, tablet PCs with full batteries will be given to student groups, and the activity will be turned into a group work. [/*][/list]

We have already defined the rational and irrational exponent of any positive real number of [i]a[/i]. Stating this, in case [i]a[/i] is a positive real number and [i]a[/i] ≠ 1, we call the function of  [i]f : IR → IR, f(x) =[math]a^x[/math][/i][img width=17,height=22]file:///C:/Users/admin/AppData/Local/Temp/msohtmlclip1/01/clip_image002.png[/img] as exponential function. [br][br]The domain of the function  [i]f(x) =[math]a^x[/math] [/i]is real numbers. We have already come up with different exponential functions by changing [i]a[/i]. [br]For instance, the functions [math]2^x[/math], [math]\frac{1}{5}^x[/math] or [math]\frac{1}{50}^x[/math] are exponential functions.[br]Let’s now find out the values of the function [math]3^x[/math] for  the x values x= -4, -3, -1, 0, 1, 2 and remember how we could draw its graph. [br]

When [i]a[/i] and [i]b[/i] are positive real numbers and [i]a ≠ 1[/i], we have already defined the number of [math]logab[/math] (the logarithm of [i]b[/i] in relation to the base of [i]a[/i])[br][br]Let’s now remember: It is equal to [i]c [/i]and the exponent of [math]ac[/math] is equal to [i]b[/i]. Saying this, we have the value of ……….. . We have the function  ………… when we replace [i]b[/i] with [i]x [/i]in the definition of the logarithm of [i]b[/i] in relation to the base of [i]a[/i]. We call this function as logarithmic function with a base.[br][br]The domain of logarithmic function is all positive real numbers. The logarithmic function could also be defined as the inverse function of exponential functions. We have already stated that exponential functions are bijective. Depending on this, if we find [i]x[/i], we will have. As we have explained in the definition of a inverse function, if we replace [i]x[/i] with [i]y[/i] and [i]y [/i]with x, we will end up with the logarithmic function, which is[br]the inverse function of [i]y[/i][i].[/i] Now, let’s examine the graph, which is the inverse function, for the values of [i]a>1 , 0<a<1[/i] and [i]a<0 [/i]by comparing them.

[b]Question 1)[/b] How is the graph for [i]a<0? What is your opinion about the possible[br]reason behind it? (Clue: What is the range  for exponential functions?)[/i][br][br][b][b]Question[/b] 2)[/b] [i]How does the graphic change for [i]0<a<1[/i] and[i] a>1?[/i][br][br][b][b][b]Question[/b][/b] 3)[/b] What could you say about the increase and decrease in the logarithmic function? For which values of a is it increasing and decreasing? [br][br]Let’s now observe the changes in the function [math]loga\left(x+b\right)+c[/math] for the values [i]a, b[/i], and [i]c.[/i][/i]

[b]Task[/b]: Try to draw the graphs of [math]log_2x[/math] and [math]log_{\frac{1}{2}}x[/math] with the help of your tablet PCs.

[b]How did you implement your lesson plan? [/b][br][br]In two lessons, I applied it comfortably in general terms, adhering to the plan. I followed the tasks I gave for graphic drawings and frequently asked questions.[br][br][b][br]Do you think you could integrate technology in class properly? [/b][br][br]I had no difficulty in implementing the program. That the students had a chance to observe the graphs in an interactive way helped reinforce their learning. [br][br][b][br]Do you think your students accomplished the outcomes of the lesson? [/b][br][br]Yes, I think they did. [br]