I saw several specific questions on using logarithm on an inequality. In this question I want a general answer. This idea came from reading Paradoxes and Sophisms in Calculus. Relevant to this topic are questions 11,12 and 13 of chapter 3 in the book. For the purpose of the question logarithm is assumed to be taken on real number. Any suggestion to define the question is welcome as I am not myself sure what other possibilities are there other than those mentioned in the questions above.
To illustrate the problem here is a suggestion:
$$ 3 \gt 2 $$ $$ => \log_x 3 > \log_x 2 $$
The question is which values of $x$ are applicable here. Also how the values of $x$ change when we change the inequality in any way like replacing one side with a function or changing values for other range like between $-1$ to $1$.
Hint
Making the problem more general, under which condition does hold the inequality $$ A=\log_x(a) - \log_x(b) >0$$ Converting to natural logarithms, this write $$\frac{\log (a)}{\log (x)}-\frac{\log (b)}{\log (x)}=\frac{\log(\frac a b)}{\log(x)}$$ which has the same sign as $${\log(\frac a b)}\,{\log(x)}$$
I am sure that you can take it from here.