Laws of Logarithms

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

\log_b(x+y) = \log_b x + \log_b y.

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 \log_b(x+y) somehow simplfies as \log_b x + \log_b y.

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 \log_b distributes just as if x+y was being multiplied by a constant. Other mistakes in this vein include  \sqrt{x+y} = \sqrt{x} + \sqrt{y}  and  (x+y)^2 = x^2 + y^2.

Along the same lines, other classic blunders are thinking that

\left(\log_b x\right)^n  simplifies as  \log_b \left(x^n \right)

and that

\displaystyle \frac{\log_b x}{\log_b y}  simplifies as  \log_b \left( \frac{x}{y} \right).

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.

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