Laws of Logarithms

One of the most common student mistakes with logarithms is thinking that

.

When I first started my career, I referred to this as the Third Classic Blunder. The first classic blunder, of course, is getting into a major land war in Asia. The second classic blunder is getting into a battle of wits with a Sicilian when death is on the line. And the third classic blunder is thinking that somehow simplfies as .

Sadly, as the years pass, fewer and fewer students immediately get the cultural reference. On the bright side, it’s also an opportunity to introduce a new generation to one of the great cinematic masterpieces of all time.

One of my colleagues calls this mistake the Universal Distributive Law, where the distributes just as if was being multiplied by a constant. Other mistakes in this vein include    and  .

Along the same lines, other classic blunders are thinking that

  simplifies as 

and that

  simplifies as  .

I’m continually amazed at the number of good students who intellectually know that the above equations are false but panic and use them when solving a problem.