A scatter plot of y against x may give hints on which transformation to use to transform the plot to a straight line for regression purposes. For example,
the figure on the left (convex increasing) suggests an higher power of $x$ e.g. $$y=\beta_0 + \beta_1w$$ with $w = x^2$ from the original $y=\beta_0+\beta_1x$,
or a lower power of y e.g. $$z=\beta_0 + \beta_1x$$ with $z=\log(y)$ from the original $y=\beta_0+\beta_1x$.
My Prof then explains that in the context of transformations, $\log(y)$ is treated as $y^0$ and hence a "lower power". Why?
If the initial data of $n$ points $\quad (x_1,y_1),(x_2,y_2),...,(x_k,y_k),...,(x_n,y_n)\quad$ don't looks like around a straight line when drawn on a graph, one can try to transform it.
For example in computing $\quad Y_k=f(y_k)\quad $ with a given function $f$. For example $f(y)=\sqrt{y}$ or $f(y)=\ln(y)$ or $f(y)=y^2$ or $f(y)=\exp(y)$ or other functions. For each one of these examples, one get a transform data : $$(x_1,Y_1),(x_2,Y_2),...,(x_k,Y_k),...,(x_n,Y_n)$$ When drawn on a graph, if it looks like around a straight line, a linear regression leads to an approximate equation of the straight line : $$Y(x)\simeq \beta_0+\beta_1 x$$ Since $Y=f(y)$ , the inverse function is $y=f^{-1}(Y)$ which leads to the function fitted to the initial data : $$y(x)\simeq f^{-1}(\beta_0+\beta_1 x)$$ For example, if the particular function $Y=\ln(y)$ allows a linear regression to the transform data, since the inverse function is $y=\exp(Y)$, the result is $$y(x)\simeq \exp(\beta_0+\beta_1 x)$$ which is an approximate function fitted to the original data.
Of course, instead of a systematic trial of many functions $f$, by inspection and considering the respective curvatures, one can chose only a few convenient functions to try, in order to save time.
Similar transformation can be done with change of variable $X_k=g(x_k)$ and trying convenient function $g(x)$ chosen in considering the respective curvatures. No need to repeat the above explanation.