Although addition and multiplication are commutative, exponentiation is not: swapping the value in the base with the value in the exponent will produce a different result (unless, of course, they are the same value):
Therefore, two different inverse functions are needed to solve equations that involve exponential expressions:
– roots, to undo exponents
– logarithms, to undo bases
Just as there are many versions of the addition function (one for each number you might wish to add), and many versions of the “logarithm” function (each with a different base), there are many versions of the “root” function: one for each exponent value to be undone.
The symbol for a root is , and is referred to as a “radical“. It consists of a sort of check mark on the left, followed by a horizontal line, called a “vinculum”, that serves as a grouping symbol (like parentheses) to Continue reading Roots and Rational Exponents: a summary
Unlike the two most “friendly” arithmetic operations, addition and multiplication, exponentiation is not commutative. You will get a different result if you swap the value in the base with the one in the exponent (unless, of course, they are the same value):
The most significant impact of this lack of commutativity arises when you need to solve an equation that involves exponentiation: two different inverse functions are needed, one to undo the exponent (a root), and a different one to undo the base (a logarithm).
Just as there are many versions of the addition function (adding 2, adding 5, adding 7.23, etc.), and many versions of the “root” function (square roots, cube roots, etc.), there are also many versions of the “logarithm” function. Each version has a “base”, which corresponds to the base of its inverse exponential expression.
Inverse Functions: Logarithms & Exponentials
Logarithms are labelled with a number that corresponds to the base of the exponential that they undo. For example, the Continue reading Logarithms
A few more interactive GeoGebra applets have been added to my collection.
Each of these graphs the function indicated in the name, with parameters that you can adjust using sliders. As you move a slider, you can watch how that parameter affects the graph of the function, and see what the resulting function definition looks like.
Each of these pages also has a set of questions following the grapher that are intended to lead students through the process of playing with and considering the effect of each constant on the graph (without giving away too much, hopefully…).
One of the hardest questions for many math teachers to answer in a way that is relevant to students is: “why do I need to know this?” “For the next course you take”, the easiest answer in many cases, does not answer the question that was usually being asked. My answers to this question obviously depend on the topic being studied at moment, and I don’t have “good” answers for all topics… but here is my list of key life skills I learned directly or indirectly from math class, with some examples of situations where I find them indispensable.
Continue reading Life Skills Learned In Math Class