It's a double edged sword. I loved math because I liked physics, and calculus text books had plenty of physics examples.

Then I tutored people doing "business math" - calculus but using finance for all its applications. Most people struggled not because of the math but because of the difficulty in understanding the finance part. And it was hard for me to give help because I had to learn the financial aspects to help them with their HW.

If the application doesn't click with you, then you now have two problems.

Exactly - the applicability has to mean something for the person. It could require several regressions before you get to a core interest of any specific student.

I never understood people that questioned the practical applicability of math. Surely they must realize that these things are so elementary that there is no way there was no practical use for them. Logarithms specifically allow you to multiply values quickly by looking up their logarithm in a table, add the values and do a reverse lookup in the same table. This is the foundation of slide rules, which is the way engineers did calculations before computers were common.

Of course we all know this, so you’re preaching to the choir. However I think it would be quite demoralizing to tell a young student who is struggling with logarithms (to follow your example) that “it’s so elementary that there must be a practical use!” Indeed, middle and high school math is, to many, the most intellectually challenging thing they’ve ever had to deal with at the time these classes are taken. Considering they haven’t even taken higher order maths yet, students likely don’t have the context to realize that “elementary ≠ easy to understand”

I’ve taught at a number of “nerd camps”, generally in the fields of Gamedev and 3D modeling. I found that, more than anything else, showing young students what they can use a tool directly for triggers the imagination and inspires them to engage further with the field. These students were able to produce Google Cardboard VR content, which at the time was impressively close to the state of the art.

Anecdote: I’ve personally seen a 10-12 year old student tinker with and research the shadow settings in Unity3D [1], which is quite complicated! And she was willing and excited to put in the work because she could see with her eyes how the options affected the shadows directly. She wanted to match the shadows with a drawing she brought in.

Too bad Logarithms are not as engaging to look at as video games...

[1] https://docs.unity3d.com/Manual/ShadowOverview.html

I have never once in my life used logarithms in that fashion. I've used them in other ways, but why would you do it that way instead of just hitting the "*" button?

Because you did not have a * button?

I this sums up my first math experience in the US high school. Learning before moving to the US, we were not allowed any calculators. My first class the teach wrote something basic like quadratic equation to solve. To my shock, out went the TI-83 graphing calculators.

I was done with the problem before my peers even hit the right keys in their pocket computers (TI-83 is not a calculator, but a computer pretending to be a calculator).

I still think the non-calculator way of teaching math gave me better skills down the road.

You don't have a calculator, so you're calculating logarithms by hand?

You understand that slide rules existed before calculators, yeah?

They are practical only to the one using them. I don't care about fluid dynamics when I stir my coffee with my well-designed spoon.

Perhaps the problem is that there is no difference between 2x + 1 = 0 and 3x + 3 = 0, they are the same kind of question but in my school days we have to do all these similar tasks for ... whatever reasons.

Then again, practice is a really good way to solidify one's understanding of a concept so its a hard thing to balance.

I only learned about this application of logarithms years after learning the definition. Usually the applications are not obvious to students and this goes on even at the university level.

I sympathise with you. It certainly seems that those people who do best at mathematics in school are just those who are best at abstract manipulation of symbols.

The example you give is an interesting one. Logarithms are quite hard to motivate[+]. Once they could be introduced as analogous to a slide rule as something to speed up multiplication (indeed that was their actual original motivation). But that isn’t convincing these days. Perhaps a motivation could be found in science (something about gradients and log-log plots). The problem is that the real reason you learn about logs is because they are related to exponential and exponential are important. Consider the differential equation:

  dy/dx = k y

This kind of equation comes up a lot in modelling things in science and other parts of the real world (eg compound interest). We can solve it:

  k dx/dy = 1/y
  k x = log(y) + C
  log(y) = k x - C
  y = exp(k x - C)
  y = A exp(k x) (where A = exp(-C) is chosen to fit initial conditions)

But this motivation isn’t much use when one doesn’t have differential equations or calculus. When I was in school I think differential equations where only in A-level further maths (age 16-18). I’m not sure whether we had logarithms at GCSE (14-16) but I think they were in A level maths which included calculus up to integration by parts and integrals/derivatives of various trig functions.

[+] By motivate I mean “put into context of some practical application”.

In the “real world” such mathematical things aren’t necessarily well motivated either. It may be that one has to calculate certain weird things because that is what clients/customers/taxes/regulations/circumstances dictate.

In applied mathematics things aren’t so well motivated. Either “we see these equations in physics sometimes” or for fluids you mumble something about boats or planes. In pure mathematics the motivation basically always boils down to “we need this crazy thing to solve a certain problem to tell if two objects are the same/different/somehow interesting because we are interested in this sort of object and in particular how to classify the different kinds of it.