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.

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