Exponential and Logarithmic Functions Essay Essay Example

Exponential and Logarithmic Functions Essay Paper This paper presents briefly the definitions. illustrations. features. and applications of both exponential and logarithmic maps. Exponential Functions The basic exponential map is denoted by degree Fahrenheit ( x ) = bx. where b. the base is a positive existent figure ( b gt ; 0 ) and b? 1. and x. the advocate. is any existent figure. ( Marcus. 2008 ; Roberts. 2008a ; Stapel. 2008a ) . Figure 1 nowadayss three illustrations of exponential graphs of the signifiers f ( ten ) = 2x. degree Fahrenheit ( x ) = 8x. and f ( ten ) = 0. 5x ( Roberts. 2008a ) . The features of the three graphs are summarized as follows: ( 1 ) the sphere of degree Fahrenheit ( ten ) is all existent Numberss ; ( 2 ) the scope of degree Fahrenheit ( ten ) is all positive existent Numberss ; ( 3 ) when B gt ; 1. the graph and its incline addition when ten additions ; ( 4 ) when 0 lt ; b lt ; 1. the graph and its incline lessening when ten decreases ; ( 5 ) the graph passes through y-axis at ( 0. 1 ) —all exponential graphs of the signifier degree Fahrenheit ( x ) = bx cross the y-axis at ( 0. 1 ) when B gt ; 0 ; ( 6 ) the graph is asymptotic to the x-axis—it does non traverse the x-axis or touch it. but it gets closer and closer to the x-axis as ten gets smaller and smaller ( Roberts. 2008a ; Stapel. 2008a ) . We will write a custom essay sample on Exponential and Logarithmic Functions Essay specifically for you for only $16.38 $13.9/page Order now We will write a custom essay sample on Exponential and Logarithmic Functions Essay specifically for you FOR ONLY $16.38 $13.9/page Hire Writer We will write a custom essay sample on Exponential and Logarithmic Functions Essay specifically for you FOR ONLY $16.38 $13.9/page Hire Writer Exponential maps are utile in existent universe state of affairss. Common applications include analysing population growing. calculating compound involvement. C dating artefacts. finding clip of decease. and ciphering exponential decay. To demo an application utilizing an exponential map. see the recognition cards with involvement compounded daily. The map depicting the balance ( chief + involvement ) on an mean day-to-day balance of $ 500 at involvement rate of 15 % will be: C ( T ) = 500 ( 1 + ( . 15/365 ) ) 365t. where t represents the figure of yearss for ciphering the involvement ( Taylor. 2008 ) . Logarithmic Functions The basic logarithmic map is denoted by degree Fahrenheit ( x ) = logb ( x ) . where b. the base. is a positive existent figure ( b gt ; 0 ) and b? 1. and x. the advocate. is any positive existent figure. The map degree Fahrenheit ( x ) = logb ( x ) is the opposite of the exponential map degree Fahrenheit ( x ) = bx. or y = logb ( x ) is tantamount to x = B Y ( Roberts. 2008b ; Stapel. 2008b ) . Figure 2 nowadayss three illustrations of logarithmic graphs of the signifiers y = log ( x ) . y = log ( x ) /log ( 2 ) . and y = log ( x ) /log ( . 5 ) ( Roberts. 2008b ) . The features of the three graphs are summarized as follows: ( 1 ) the sphere of degree Fahrenheit ( ten ) is all positive existent Numberss ; ( 2 ) the scope of degree Fahrenheit ( ten ) is all existent Numberss ; ( 3 ) when B gt ; 1. the graph increases when ten additions ; ( 4 ) when 0 lt ; b lt ; 1. the graph decreases when ten lessenings ; ( 5 ) the graph passes through x-axis at ( 1. 0 ) —all exponential graphs of the signifier degree Fahrenheit ( x ) = logb ( x ) cross the x-axis at ( 1. 0 ) when B gt ; 0 ; ( 6 ) the graph is asymptotic to the y-axis—it does non traverse the y-axis or touch it. but it gets closer and closer to the y-axis as y gets smaller and smaller ( Roberts. 2008b ; Stapel. 2008b ) . Logarithmic maps are similarly utile in existent universe state of affairss. such as in gauging the figure of species. finding the magnitude of temblors or the strength of sound moving ridges. finding the sourness of a solution. and ciphering the rate of bring forthing fresh H2O from salt H2O ( Saye. 2008 ) . To demo an application in utilizing logarithmic map. see a solution’s pH is defined by P ( T ) = -log10 ( T ) . where T is the hydronium ion concentration in moles per litre. The map for happening the pH of a solution with hydronium ion concentration 4. 5 ten 10-5 will be p ( T ) = -log10 ( 4. 5 ten 10-5 ) ( The University of Iowa. 2006 ) . Decisions The exponential and logarithmic maps have inverse relationships. The illustrations illustrate their graphical features and demo their reverse relationships. Both have utile applications in existent state of affairss. like analysing population growing and depletion rate of species. Mentions Marcus. Nancy. ( 2008 ) . Exponential Functions. S. O. S. Mathematics Home Page. Retrieved May 20. 2008. from hypertext transfer protocol: //www. sosmath. com/algebra/logs/log4/log4. hypertext markup language. Roberts. Donna. ( 2008a ) . Exponential Functions. Algebra 2/Trig Lesson Page. Oswego City School District Regents Exam Prep Center. Retrieved May 21. 2008. from hypertext transfer protocol: //www. regentsprep. org/Regents/math/algtrig/ATP8b/exponentialFunction. htm. Roberts. Donna. ( 2008b ) . Logarithmic Functions. Algebra 2/Trig Lesson Page. Oswego City School District Regents Exam Prep Center. Retrieved May 21. 2008. from hypertext transfer protocol: //www. regentsprep. org/Regents/math/algtrig/ATP8b/logFunction. htm. Saye. D. ( 2008 ) . Word Problems: Logarithmic Models. Algebra Lab. Mainland High School. Retrieved May 21. 2008. from hypertext transfer protocol: //www. algebralab. org/Word/Word. aspx? file=Algebra_LogarithmicModels. xml. Stapel. Elizabeth. ( 2008a ) . Exponential Functions: Introduction. Purplemath. Retrieved May 20. 2008. from hypertext transfer protocol: //www. purplemath. com/modules/expofcns. htm. Stapel. Elizabeth. ( 2008b ) . Logarithm: Introduction to the Relationship. Purplemath. Retrieved May 20. 2008. from hypertext transfer protocol: //www. purplemath. com/modules/logs. htm. Taylor. S. ( 2008 ) . Applications of Exponential Functions. Algebra Lab. Mainland High School. Retrieved May 21. 2008. from hypertext transfer protocol: //www. algebralab. org/lessons/lesson. aspx? file=Algebra_ExponentsApps. xml. The University of Iowa. ( 2006 ) . Real-World Applications of Logarithmic Function. Mathematics Matters at IOWA. Retrieved May 21. 2008. from hypertext transfer protocol: //www. uiowa. edu/~examserv/mathm atters/tutorial_quiz/log_exp/realworldappslogarithm. hypertext markup language.

How to Write a Letter of Complaint

How to Write a Letter of Complaint Heres a project that will introduce you to brainstorming and give you practice in group writing. You will join with three or four other writers to compose a letter of complaint (also called a claim letter). Consider Different Topics The best topic for this assignment will be one that you and the other members of your group truly care about. You may write to the dining hall supervisor to complain about the quality of the food, to an instructor to complain about his or her grading policies, to the governor to complain about cuts to the education budgetwhatever subject the members of your group find interesting and worthwhile. Begin by suggesting topics, and ask one member of the group to write them down as they are given. Dont stop at this point to discuss or evaluate the topics: simply prepare a long list of possibilities. Choose a Topic and Brainstorm Once you have filled a page with topics, you can decide among yourselves which one you would like to write about. Then discuss the points that you think should be raised in the letter. Again, have one member of the group keep track of these suggestions. Your letter will need to explain the problem clearly and show why your complaint should be taken seriously. At this stage, you may discover that you need to gather additional information to develop your ideas effectively. If so, ask one or two members of the group to conduct some basic research and bring their findings back to the group. Draft and Revise a Letter After collecting sufficient material for your letter of complaint, elect one member to compose a rough draft. When this has been completed, the draft should be read aloud so that all members of the group can recommend ways to improve it through revision. Each group member should have the opportunity to revise the letter according to the suggestions made by the others. To guide your revision, you may want to study the structure of the sample complaint letter that follows. Notice that the letter has three distinct parts: An introduction that clearly identifies the subject of the complaint.A body paragraph that (a) clearly and specifically explains the nature of the complaint, and (b) provides the reader with all of the information needed to provide an appropriate response.A conclusion that clearly states what actions are needed to remedy the problem. Annie Jolly110-C Woodhouse LaneSavannah, Georgia 31419November 1, 2007Mr. Frederick Rozco, PresidentRozco Corporation14641 Peachtree BoulevardAtlanta, Georgia 303030Dear Mr. Rozco:On October 15, 2007, in response to a special television offer, I ordered a Tressel Toaster from your company. The product arrived in the mail, apparently undamaged, on October 22. However, when I tried to operate the Tressel Toaster that same evening, I was distressed to find that it did not fulfill your claim to provide fast, safe, professional hair-styling. Instead, it severely damaged my hair.After following the instructions to set up the toaster away from other appliances on a dry counter in my bathroom, I inserted the steel comb and waited 60 seconds. Then I removed the comb from the toaster and, following the instructions for a Venusian Curl, ran the hot comb through my hair. After just a few seconds, however, I smelled burning hair, and so I immediately placed the comb back into the toaster. When I did this, sparks flew from the outlet. I reached to unplug the toaster, but I was too late: a fuse had already blown out. A few minutes later, after replacing the fuse, I looked in the mirror and saw that my hair had been scorched in several spots.I am returning the Tressel Toaster (along with the unopened bottle of Un-Do Shampoo), and I expect a full refund of $39.95, plus $5.90 for shipping costs. In addition, I am enclosing a receipt for the wig I purchased and will have to wear until the damaged hair grows out. Please send me a check for $303.67 to cover the refund for the Tressel Toaster and the cost of the wig.Sincerely,Annie Jolly Notice how the writer has delivered her complaint with facts rather than emotions. The letter is firm and direct but also respectful and polite. Revise, Edit, and Proofread Your Letter Invite one member of your group to read aloud your letter of complaint and respond to it as if he or she had just received it in the mail. Does the complaint sound valid and worth taking seriously? If so, ask the members of the group to revise, edit, and proofread the letter one final time, using the following checklist as a guide: Does your letter follow the standard format shown in the example above?Does your letter consist of an introduction, a body paragraph, and a conclusion?Does your introductory paragraph clearly identify what you are complaining about?Does your body paragraph clearly and specifically explain the nature of the complaint?In the body paragraph, have you provided the reader with all of the information needed if he or she is to respond effectively to your complaint?Have you conveyed your complaint calmly and clearly, relying on facts rather than emotions?Have you clearly organized the information in your body paragraph so that one sentence leads logically to the next?In your conclusion, have you clearly stated what action(s) you want your reader to take?Have you proofread the letter carefully?