Innovative Teaching Technique for the Exponential and Logarithmic Functions

Abstract Innovative Teaching Technique for the Exponential and Logarithmic Functions Making engineering education more understandable to students can be difficult due to the demanding mathematical requirements the major demands. This is especially true for mathematically under prepared students. One specific area of great difficulty for under prepared students is understanding the logarithmic functions. Part of the problem is that the logarithmic functions seem mysterious to them because students know them as only two keys on a calculator "ln" and "log" The logarithmic functions are classified as transcendental functions. A transcendental function cannot be written as a finite combination of algebraic expressions. The key word is FINITE. This fact in most cases eliminates the equation form ever being seen by students. This paper does not aim to eliminate the above-mentioned calculator keys from calculations but wishes to have students use the actual equation for a few examples. To help take the mystery out of the calculator "ln" key, a few problems will contain the use of the actual transcendental function where students only use a calculator to determine the first three terms in the logarithm Power Series Expansion formula. For simplicity only the first three terms in the series will be used. The first three terms still require a lot of work. These values will then be compared to a value obtained with the "ln" calculator key. Function notation will also be addressed. Most students don't even realize that the familiar f in f(x) is being replaced by "ln". In addition, this paper is not written from a research perspective. There was no collected student data from surveys as to the effectiveness of this supplemental chapter. This paper will contain the fully written abbreviated supplemental chapter needed to be included in any pre-calculus course. This chapter will start with the introduction of the exponential functions and will show the transition to the inverse logarithmic functions. In particular, the traditional notation for inverse functions will not be used. The traditional notation loses the fact that the dependent y value now becomes the independent variable. The exponential function is in the form of y = f(x) = b to the power of x. The inverse notation will be represented as x = f(y) = logb(y), not the traditional y = f-1(x). In addition, the base will only be represented as b, it will not be replaced with x or any other variable when solving exponential equations. This supplemental chapter aims to reinforce the identity of the variables x, y, and b and not to interchange them. Both the exponential function with base e, y = f(x) = e to the power of x, and the natural logarithm x = f(y) = loge(y) =ln(y) will be graphed. The latter will be plotted with x on the vertical axis and y on the horizontal axis.