I’m currently taking a “Mathematics for Teaching Algebra” class, and today we discussed logarithms. The teacher gave us a problem I’d like to share. Also, the class discussion causes me to come up with the following word problems which give motivation for wanting an inverse exponential function. These are straight-forward to solve with logs and give all the information needed to solve them in the problem (not WCYDWT), but they are reasonable questions to ask.
The first question I thought of was “Billy has $100 and wants to buy a skateboard that costs $150. His money is in saving accounts earning 3% interest per year, compounded continuously. How long will he have to wait to buy the skateboard using only his savings and earned interested?” This is nice as it is a problem that students can relate to and care about, but unfortunately, the answer is essentially “way too long, just get a job already.”
The next sample question I came up with is in biology, which I am less intimately familiar with than bank accounts. “Alex and Beth are growing a fungus sample for their biology project. They have 5 grams of fungus and need 20 grams to do the analysis they want to. If the fungus increases its weight by 10% each day, how long will they have to wait before they can do their analysis?”
This problem requires some understand of combinitorics and probability to understand, but is closely related to real computer science applications. “I have two button device that I want to lock with a password. The password will be a fixed number of button presses of one or the other button. How many button presses does the password have to be to assure that someone who knew the length of the password would have less than a one in a million chance of guessing correctly on their first try?”
All these problem suffer the inherent problem with word problems, which is that they test language skills as much as math skills, but that is a rant for another day. Other than that, what do you think of these?
On the other hand, the following problem presented by my teacher has no real world connection that I can think of, but is mathematically elegant, and a wonderful test of depth of understand of logarithms. I like this problem better than all the ones I made up.
(Equation made with LaTeX online here. I should possibly find a more streamlined way of making equations than using a website.)