Our pre-calculus class was going over how to combine sums and differences of logarithm expressions into a single logarithm when Steven Orton (He gave permission to use his name.) asked, “What about PEMDAS?” PEMDAS is an acronym used for remembering the order of operations of evaluating algebraic expressions. I muttered a few sentences, said I probably didn’t understand the question, and offered to discuss it after class.
I woke up a 1:30 am the next morning knowing what Steven meant. At the beginning of the next class I warned my students to work left to right when combining logarithms and meet with Steven after class with the following explanation.
First let’s revisit the original issues remembering this codicil to PEMDAS :
Addition/subtraction are on the same hierarchical level of the order of operations and must be worked left to right. The same holds for multiplication/division.
Here is the issue for addition/subtraction.
Addition-Subtraction Order of Operations
The final numbers differ. Our agreement is that the first process, moving left to right, gives the “correct” answer. Correct in this context means what we have agreed such expressions should be evaluated left to right since order makes a difference.
Such expressions can be made unambiguous by remembering the definition of subtraction is just adding the opposite. Thus,
How to “Fix” Addition – Subtraction
Now any order of evaluation gives the same result since addition is associative.
The same argument works for multiplication/division.
Multiplication-Division Order of Operations
The first evaluation is “correct” and we can “fix” the issue by remembering that division is just multiplying by the reciprocal. Thus,
How to “Fix” Multiplication-Division
Now any order of evaluation gives the same result since multiplication is associative.
Now let’s use the properties of logarithms to simplify the following.
Logarithm Simplification Order of Operations
Again left to right gives the desired answer. This can be “fixed” as follows.
How to “Fix” Logarithm Simplification
This simplification uses a property of logarithms that we didn’t make explicit in class, namely,
A Property of Logarithms
We have an interesting application of PEMDAS for simplifying logarithm expressions addressing about both subtraction and division.
All of the above to fix (explain) a defect in standard algebraic expression notation. Thanks to Steven for the excellent question.