Logarithms are not hard to work with when only one base is involved (as in most real-life problems); but they can be challenging when each log has a different base. Here, we’ll look at a few problems in which we have to compare logarithms with different bases, showing various strategies.
(A new question of the week) Looking for a new topic, I realized that a recent question involves determinants, and an older one provides the background for that. We’ll continue the series on determinants by seeing how they can be used in finding the inverse of a matrix, and how something called the adjugate matrix …
A popular kind of word problem tells us how many people (or cats, or hens, …) it takes to make some number of houses (or kill some number of mice, or lay some number of eggs) in some amount of time, and then asks us to fill in one of the blanks for a different …
How Many A’s Can Make This Many B’s in This Much Time? Read More »
(A new question of the week) We are often asked to help a student understand a solution to a problem, obtained from a book or a website, that is not fully explained there. Here, we’ll look at a rather odd demonstration that a function satisfies a differential equation, both figuring out what the author did, …
(A new question of the week) This week, we’ll look at two recent questions about how parentheses (brackets) are used, how they relate to the properties we use in algebra that let us add or drop them, and the related concept of factoring a polynomial. They are examples of how student questions can touch on …
Parentheses and the Associative and Distributive Properties Read More »
(A new question of the week) Extraneous roots can not only confuse the final solution to a problem; they can also make it harder to solve in the first place if you don’t deal with them early. Here is a relatively complicated rational equation, two questions about its solutions, and several ways to make it …
A Rational Equation, With and Without Extraneous Roots Read More »
(A new question of the week) Some problems can be done either by algebra or by basic arithmetic methods and some creativity; and although algebra generally makes work easier by making it routine, sometimes special-purpose thinking (once you have thought it!) can be quicker. Here we have a problem where a creative method didn’t quite …
Last week we looked at a set of special rules for working with proportions, which have been largely replaced by the more general “tool” of algebra (the “Swiss army knife” of problem solving, which can do the job of many specialized tools), though the latter can still be useful. We still find that many students …
A new question of the week A simple question about the Basic Proportionality Theorem (which goes by several other names, including Thales’ Theorem) and its relationship to similar triangles leads to some helpful ideas about how to choose a suitable manipulation at each step in a proof, a skill central to good problem solving. In …
Last time, looking at degenerate polygons, I mentioned some other issues pertaining to the definition of a polygon. Let’s take the opportunity to look at them. This post supplements what was said previously in What is a Polyhedron … Really?