Exponential and Logarithmic Functions

This sample of an academic paper on Exponential Essay reveals arguments and important aspects of this topic. Read this essay’s introduction, body paragraphs and the conclusion below.

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 ) .

 Exponential Maps

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.

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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/mathmatters/tutorial_quiz/log_exp/realworldappslogarithm. hypertext markup language.