# 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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