# Log rules!

This is the first in my series of posts that describes lesser known tricks that I use to teach specific mathematical topics. The content about which I’ll be blogging will range from elementary school level through calculus; hopefully, there is a little something for everybody. When the muse strikes, I might even embellish the more elementary topics by giving a little extra insight from the advanced point of view. So, without further ado…

# Tricks of the Trade

(with Professor Glesser)

It must be strange to teach logarithms to students now if you were teaching in 1960’s or 1970’s. For then, there was an application you could immediately point to: logarithms make multiplication and division easier by turning them into addition. As students didn’t own calculators, only slide rules, as long as they bought into the usefulness of multiplication, you were solid. However, today, we have to rely on less obvious applications (e.g., logarithmic scales, being the inverse of the exponential function, being an antiderivative for $\dfrac{1}{x}$). One of the drawbacks is that the students still need to learn the basic rules of logarithms but without the immediate motivation. [Update: A very nice worksheet to help this initial process in given by Kate Nowak here.] $\begin{array}{lll} \log(x^n) & = & n\log(x) \\ \log(xy) & = & \log(x) + \log(y) \\ \log(\dfrac{x}{y}) & = & \log(x) - \log(y) \end{array}$

(When I use the symbol $\log$ it refers to any base.)

Of course, these rules are why logarithms were so important for computational purposes. But, without getting a chance to use these rules a few hundred times, I’ve noticed students have difficulty remembering them. One technique I’ve used in the past with success is to tie it to something they do know.

Enter PEMDAS

Nearly all students are familiar with this acronym and many know mnemonics for remembering it (e.g., Please Excuse My Dear Aunt Sally). In case you don’t, the letters stand for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction and the order refers to the order in which you should evaluate an expression. However, it is also a keen way of recalling the logarithm rules. First, I always encourage my students to write PEMDAS as follows: $\begin{array}{ll} P & \\ E & \\ M & D \\ A & S \end{array}$

as it helps reinforce the idea that multiplication and division as well as addition and subtraction are related operations. Next, to get the rules for logarithms, simply draw down arrows as follows: $\begin{array}{ll} P & \\ E & \\ \downarrow & \\ M & D \\ \downarrow & \downarrow \\ A & S \end{array}$

The arrows tell you that exponentiation turns into multiplication (rule 1 above), multiplication turns into addition (rule 2) and division turns into subtraction (rule 3). Invariably, a student will say, “but my problem is remembering which way to write it.” They mean, for instance, which of the following is the correct rule 2? $\log(xy) = \log(x) + \log(y)$

or $\log(x)\log(y) = \log(x+y)$

This is where the P comes in (note that we didn’t use it above). The P, which still stands for parentheses, tells you that you what you are changing goes in the parentheses. So, if you are turning multiplication into addition, the multiplication is what goes in the parentheses. Occasionally, students still have problems remembering and I tell them to look at what is closer to the P. In the diagram, since M is closer than A to P, the multiplication goes inside the parentheses.

Next time

In the next edition of Tricks of the Trade, I’ll discuss adding and subtracting fractions…in the quotient field of an integral domain.

Cheers.

## 4 thoughts on “Log rules!”

1. Allen Glesser

Great stuff Adam! I’ll subscribe . . .

2. Fred Flarben

Bookmarked! Thanks for posting this. I’ve frequently had trouble with logs because my mind always wants to convert everything to exponents to do any math with them. My problem is that I do fully understand what logs represent, but the mechanics — the mental memory of figuring with them — always wants to switch to exponents because exponents seem intuitive and natural and logs seem backwards. This little reworking of the PEMDAS technique is no more “harmful” than remembering the steps involved in long division or figuring out what power of 10 to use in hand calculations involving scientific notation, or SOHCAHTOA. These are shortcuts to figuring…they don’t mean that one doesn’t understand what is going on.