Time for another rant. Some topics in math are ripe for abuse. Previously, I looked at the sorry state of rational functions and exponents. Another has drawn my gaze and that's logarithms. I admit that in the two previous scenarios and this one, I am totally guilty of each bad practice.
Logs are usually in an unfortunate position. Often taught towards the end of the year, and tough to demonstrate as practical. Should you go on to higher math and science there are plenty of reasons to love logs, but at the high school level it's pretty rough.
Why? Well, log laws. Logs have a handful of neat properties, and those properties were a way to simplify hard multiplication prior to calculators, and they power the slide rules that sent us to the moon. Log tables were an important part of textbooks. With computers better equipped to handle nasty numbers, the properties get turned into a "nice to know" sideshow.
Case in point:
The student is asked to extract these single logarithms into several to demonstrate their knowledge of log laws. The glaring problem is why? Why does this matter? Most of these can't be graphed, but let's examine #37.
C'mon. This sort of stuff right here, the demonstration of properties with no context is what undermines math education. Sure, they'll diligently expand the log for you, but what has that accomplished? What better understanding of logarithms do they have now that they can expand some unwieldy function into a string of unwieldy functions? Why is it a problem worth solving?
A better approach might be the algebra that proves this scenario.
Lacking real world context? Yep. Contrived? Yep. Unwieldy? Yep. But what's different? Instead of demonstrating log laws for the sake of demonstrating log laws, you can see where they expand the algebra skill set. That hey, there's two answers, so an embedded quadratic shares the properties of a normal one?
Curriculum needs to flow. The work done in September should connect to that taught in March. A series of random diversions dilutes the big picture. Log laws have a place, but they should not be trotted out to the center ring and forgotten a week later.