Exponential and Logarithmic Functions

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Exponential and Logarithmic Functions

This paper presents briefly the definitions, examples, characteristics, and applications of both exponential and logarithmic functions. Exponential Functions The basic exponential function is denoted by f(x) = bx, where b, the base is a positive real number (b > 0) and b ? 1, and x, the exponent, is any real number. (Marcus, 2008; Roberts, 2008a; Stapel, 2008a). Figure 1 presents three examples of exponential graphs of the forms f(x) = 2x, f(x) = 8x, and f(x) = 0. 5x (Roberts, 2008a).

The characteristics of the three graphs are summarized as follows: (1) the domain of f(x) is all real numbers; (2) the range of f(x) is all positive real numbers; (3) when b > 1, the graph and its slope increase when x increases; (4) when 0 < b < 1, the graph and its slope decrease when x decreases; (5) the graph passes through y-axis at (0,1)—all exponential graphs of the form f(x) = bx cross the y-axis at (0, 1) when b > 0; (6) the graph is asymptotic to the x-axis—it does not cross the x-axis or touch it, but it gets closer and closer to the x-axis as x gets smaller and smaller (Roberts, 2008a; Stapel, 2008a).

Exponential functions are useful in real world situations. Common applications include analyzing population growth, computing compound interest, carbon dating artifacts, determining time of death, and calculating exponential decay. To show an application using an exponential function, consider the credit cards with interest compounded daily. The function describing the balance (principal + interest) on an average daily balance of $500 at interest rate of 15% will be: C(t) = 500 (1 + (. 15/365))365t, where t represents the number of days for calculating the interest (Taylor, 2008).

Logarithmic Functions The basic logarithmic function is denoted by f(x) = logb(x), where b, the base, is a positive real number (b >0) and b ? 1, and x, the exponent, is any positive real number. The function f(x) = logb (x) is the inverse of the exponential function f(x) = bx, or y = logb (x) is equivalent to x = b y (Roberts, 2008b; Stapel, 2008b). Figure 2 presents three examples of logarithmic graphs of the forms y = log(x), y = log(x)/log(2), and y = log(x)/log(. 5) (Roberts, 2008b).

The characteristics of the three graphs are summarized as follows: (1) the domain of f(x) is all positive real numbers; (2) the range of f(x) is all real numbers; (3) when b > 1, the graph increases when x increases; (4) when 0 < b < 1, the graph decreases when x decreases; (5) the graph passes through x-axis at (1,0)—all exponential graphs of the form f(x) = logb(x) cross the x-axis at (1,0) when b > 0; (6) the graph is asymptotic to the y-axis—it does not cross 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 functions are likewise useful in real world situations, such as in estimating the number of species, determining the magnitude of earthquakes or the intensity of sound waves, determining the acidity of a solution, and calculating the rate of producing fresh water from salt water (Saye, 2008). To show an application in using logarithmic function, consider a solution’s pH is defined by p(t) = -log10 (t), where t is the hydronium ion concentration in moles per liter. The function for finding the pH of a solution with hydronium ion concentration 4. 5 x 10-5 will be p(t)= -log10(4. 5 x 10-5) (The University of Iowa, 2006).

Conclusions The exponential and logarithmic functions have inverse relationships. The examples illustrate their graphical characteristics and show their inverse relationships. Both have useful applications in real situations, like analyzing population growth and depletion rate of species.

References

Marcus, Nancy. (2008). Exponential Functions. S. O. S. Mathematics Home Page. Retrieved May 20, 2008, from http://www. sosmath. com/algebra/logs/log4/log4. html. Roberts, Donna. (2008a). Exponential Functions. Algebra 2/Trig Lesson Page, Oswego City School District Regents Exam Prep Center. Retrieved May 21, 2008, from http://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 http://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 http://www. algebralab. org/Word/Word. aspx? file=Algebra_LogarithmicModels. xml. Stapel, Elizabeth. (2008a). Exponential Functions: Introduction. Purplemath. Retrieved May 20, 2008, from http://www. purplemath.com/modules/expofcns. htm. Stapel, Elizabeth. (2008b). Logarithms: Introduction to the Relationship. Purplemath. Retrieved May 20, 2008, from http://www. purplemath. com/modules/logs. htm. Taylor, S. (2008). Applications of Exponential Functions. Algebra Lab, Mainland High School. Retrieved May 21, 2008, from http://www. algebralab. org/lessons/lesson. aspx? file=Algebra_ExponentsApps. xml. The University of Iowa. (2006). Real-World Applications of Logarithmic Function. Math Matters at IOWA. Retrieved May 21, 2008, from http://www. uiowa. edu/~examserv/mathmatters/tutorial_quiz/log_exp/realworldappslogarithm. html.

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  • University/College: University of California

  • Type of paper: Thesis/Dissertation Chapter

  • Date: 29 December 2016

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