I have a random real variable $x$ with pdf $f(x;a,b)=\frac{1}{x(\ln{b}-\ln{a})}$, $0<a\leq x \leq b$ (i.e. its logarithm is uniformly distributed between $a$ and $b$). This random variable is then transformed by one of the following two functions (just depending on the sign of $c$, which is a constant for the purpose of this question):
$g(t;c>0)=(c + \sqrt{t})^2$
$g(t;c\leq0)=\begin{cases} 0 & t < c^2 \\ (c + \sqrt{t})^2 & t \geq c^2 \end{cases}$
$g(t)$ is therefore always monotonously increasing or constant. I'd like to find a pdf for $g(x)$ (i.e. this random real variable transformed by $g$), but haven't got a clue where to start, particularly for the piecewise case. The bounds of the pdf will obviously be transformed by $g$, and the inequality $0 < a \leq x \leq b$ will remain true, unless $c < 0$ and $ a, b < c^2$, in which case the pdf will just be the dirac delta function.