Corollary 1

If $y$ is a function in $\mathfrak{K}$, not identically zero on a neighbourhood of $+\infty$, then $\mathfrak{K}(\log|y|)$ is a Hardy field.

Indeed, $(\log|y|)' = y'/y$ is equal to a function in $\mathfrak{K}$ on an interval $[ x_0, +\infty[$.

COROLLARY 2 If $y$ is any function in $\mathfrak{K}$, then $\mathfrak{K}(e^y)$ is a Hardy field.

Indeed, $(e^y)' = e^y y'$, and $y'$ is equal to a function in $\mathfrak{K}$ on an interval $[ x_0, +\infty[$.

COROLLARY 3 If $\mathfrak{K}$ contains the real constants, and if $y$ is a function in $\mathfrak{K}$, not identically zero on a neighbourhood of $+\infty$, then $\mathfrak{K}(|y|^{\alpha})$ is a Hardy field for every real number $\alpha$.

Indeed, $\frac{d}{dx}(|y|^{\alpha}) = |y|^{\alpha} (\alpha y'/y)$, and $\alpha y'/y$ is equal to a function in $\mathfrak{K}$ on an interval $[ x_0, +\infty[$.

Finally, we note that if $y$ is the primitive of any function in $\mathfrak{K}$ then $\mathfrak{K}(y)$ is again a Hardy field.