# Express A in terms of B

It’s a long weekend here. Too nice to do Calculus. Let’s do pre-calculus instead.

Example: Express $\displaystyle \log\frac{1}{x^2}$ in terms of $\log x$.

The textbook (in its 13th edition) proceeds thus.

$\displaystyle \log\frac{1}{x^2}=\log x^{-2}=-2\log x$. Here we have assumed that $x>0$. Although $\log(1/x^2)$ is defined for $x\ne 0$, the expressions $-2\log x$ is defined only if $x>0$. Note that we do have $\displaystyle \log\frac{1}{x^2}=\log x^{-2}=-2\log |x|$ for all $x\ne 0$.

This leaves me doubly confused. Which of two answers is a student supposed to give? And what does “express A in terms of B” really mean?

Concerning the latter, I can only suppose it means: exhibit a function $F$ such that $A=F\circ B$ (using the composition notation, since $A$ and $B$ are functions themselves). With this interpretation, the correct answer should be: it is impossible to express $\displaystyle \log\frac{1}{x^2}$ in terms of $\log x$.