Ideas from Classroom Teachers for
Exponential and Logarithmic Functions
To introduce the rules of exponents, give students the opportunity to explain why
b^0 = 1
b^n = 1/b^n
b^ 1/n = nth root of b
|Ex1:||b^n / b^n obviously = 1. By a previous rule,|
b^n / b^n = b^(n-n) = b^0, which must = 1.
|Ex2:||b^n × b^n = b^0 = 1 by previous rules.|
Solving for b^n, b^n = 1/b^n.
|Ex3:||b^(1/2) × b^(1/2) = b^1 = b by previous rules.|
By the definition of square root, b^(1/2) = √b.
To introduce logarithms, including the concept of logs as the inverse of exponentials, compare to the square root function as the inverse of the square function.
|y = √x is the inverse function of y = x^2, x>0.|
|√16 = 4 because 4^2 = 16; √81 = 9 because 9^2 = 81.|
|Likewise, y = logb(x) is the inverse of y = b^x.|
|log2(32) = 5 because 2^5 = 32; log10(1000) = 3 because 10^3 = 1000, etc.|
Point out that logs are exponents. The rules of logarithms can be proved using the definition of a logarithm and the laws of exponents. Relate rules of logarithms to rules of exponents:
|b^m × b^n = b^(m + n) ⇔ log(mn) = log m + log n.|
|b^m / b^n = b^(m n) ⇔ log(m/n) = log m log n.|
|(b^m)^r = b^(mr) ⇔ log(m)^r = r log m.|
Applications: In addition to the usual exponential applications, many of our (x, y) relationships encountered in the statistics unit fit an exponential model. The current unit provides an opportunity to revisit the statistics concepts learned earlier.
Growth and decay can be taught as a real-life problem with cooperation from the science field. Perhaps some interdisciplinary unit can be taught or some real-life growths can be measured and discussed.
Mathematical Investigations by Dale Seymour has application problems that would work very well in this section.
The "Wiggies" activity from the NCTM Mathematics Assessment: A Practical Handbook for Grades 912 is a great population growth activity. The solving of equations and work with logarithms involve practice with a traditional approach; however, if inquiry approaches or real-world applications are incorporated, it will help make the learning more authentic.
A website I recommend for exploration and review of exponential and logarithmic functions: http://www.purplemath.com/ (search for "exponential" or "logarithmic").
The natural number e has many beautiful connections in mathematics and a fascinating history. Suggested resource: e: The Story of a Number by Eli Maor.
Other interesting applications include problems involving pH, decibels as measuring sound intensity, and the Richter scale for comparing the magnitude of earthquakes.
Team teaching with the science department would enhance the understanding of growth and decay. The recent interest in forensic science, sparked by several TV shows, has gained the attention of many students.
Alternative assessment opportunities abound for this unit as connected with the sciences (biology, chemistry, and physics).
For activities I recommend Navigating Through Algebra, Grades 9-12 (especially for exponential functions) and NCTM’‘s Mathematics Teacher.
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