What can I say on scatter plots generated with log transformed variables? Please imagine we got scatter plots revealing a (1)rising straight line (/), (2)horizontal line (-), or (3)u-shape line (U), after log-transforming (a)one of two variables, or (b)both variables. So, in total, 6 plots.
I wonder what I can tell, having these plots. Can I understand them in a similar way to the way when I read plots of non-transformed variables? Considering that the transformation does not change order, I guess such approach may be right. So, for example:

*

*For 1a and 1b plot, "As x variable increases, y variable tends to increase"

*For 3a and 3b plot, "In the first half of x, as one x increases, y variable tends to decrease, but in the second half, y increases"

*For 2a and 2b plot, "There is no relationship".

Is this correct? Can you tell me what I can draw from each of the plots at least / at most?
If yes to the above, is it a good/reasonable practice to log-transform and see if there are relationships, when we do visual exploration?

Since:
$$
x < y \quad \Leftrightarrow \quad \log(x) < \log(y),\quad\quad \mbox{for all} \; x,y > 0
$$
(here, $\log$ indicates the natural logarithm) all your statements of "increasing", "decreasing", "constant", "U-shape" stay the same after a $\log$-transformation.
Transforming your data should only be done if helpful. But it is sometimes useful to do $log$-transformations. E.g., you might want to obtain a constant error if in the original data the error is proportional to the mean.
But, depending on your problem, many other transformations could be useful, too.
