Applied Math - Landlords and Logarithms
Jun 07, 2021

As a math and science tutor and owner of 24HourAnswers.com, I'm used to handling students' questions, but the one that is most difficult is: When am I ever going to need this?

Most of the time, I offer a two-part answer. The first part is an admission that much of what they are being taught they will most likely never need directly. The second part is a simple reminder that they do need it to do well in class, which ultimately has a bearing on the rest of their lives. I know it's a weak rebuttal, and I'm sure it fails more often than not to turn the tide of student opinion.

Recently, however, I stumbled upon a math skill that really saved the day, and I couldn't wait to tell my students. The story has to do with something in math called logarithms and how they helped me out of a jam as a landlord, strange as that may sound.

My tenant Jerry and his son were moving out of the house I rent to them in Maryland because they can't pay the rent anymore. Several years earlier, they had a period of insolvency, and at a certain point I was forced to have them sign a Confessed Judgment Note, a type of legal I.O.U., for the unpaid rent. A Note not only details the amount owed, but includes a payment schedule with all the specifics that a loan contract would carry. By signing such a Note, a tenant is acknowledging that he owes the money and is agreeing to pay it back. Confessed Judgment Notes are to landlords as light sabers are to Jedi Knights.

For this first Note, I knew everything but the size of Jerry's monthly payment, and I found the payout annuity formula in one of my math books. For those who are not afraid to look at it, here it is:

where m is the monthly payment, P is the amount of the loan, r is the annual interest rate (entered as a decimal), t is the time in years, and n is the number of payments per year.

Eight months ago, Jerry ran out of money again, but this time we both knew he had to find a less expensive place to live. Between the 18 payments remaining on the current Note together with the most recent months of unpaid rent, Jerry now owed me \$14,356.52.

The time had come to construct a new Confessed Judgment Note, and Jerry told me he would only be able to pay \$200 per month after moving out. I knew the monthly payment this time, but did not know the number of months he would need to make those payments. In short, I needed to solve the equation given above for the variable t.

If you are willing to go back and look at the formula once more, you'll notice that the time variable is in the exponent. Only logarithms allow you to solve that equation for time, and that is exactly what I did. (Not all equations with variables as exponents are solved immediately by application of logarithm rules. For an interesting example of this, see the pre-calculus subject page on this website.

Without logarithms, I would have been at the mercy of expensive lawyers using specialized software to do the calculations for me. To all my students, remember that knowledge really is power.

There is a more important message here for all students, not just my own. Everything that is taught in school is important and useful, but educational systems are like large, logging machines - too busy processing trees to worry about the forest. That doesn't mean teachers shouldn't make more of an effort to show the importance of what they teach, but if they were required to show relevance for every item in every curriculum, the educational process would grind to a halt. My advice to students is to realize that your education is the single most important gift you will ever receive, even when you don't see the immediate usefulness.

Lowell Parker, Ph.D.
Empire State College

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