Can someone please help me to derive pdf for $X$, $$ X = \frac{\ln(f_1) - \ln(f_2)}{b_2-b_1} $$ here $f_1$ and $f_2$ are normal distributions with different means and standard deviations, and $b_1$ and $b_2$ are constants. Ultimately, what I need is to find mean and standard deviation of $X$.
I'm not sure how this can be derived when both log and difference are involved.
A first point that is not very clear in your question:
(1) are $f_1$ and $f_2$ random variables ?
(2) or are they densities $f_i(x)=K_i exp(-(x-m_i)^2/(2 \sigma_i^2))$ ? In this case, it would mean a mixture of densities.
My answer works for case (1) and the very particular case where $f_1$ and $f_2$ are random variables following a $N(0,1)$ distribution.
$X$ can be written $k \ln(\frac{f_1}{f_2})$ where $k=1/(b_2-b_1)$ is a constant (assumed >0).
But $\frac{f_1}{f_2}$ follows a standard Cauchy distribution (http://mathworld.wolfram.com/NormalRatioDistribution.html) with
$$\text{pdf :} \ \ f_C(x)=\dfrac{1}{\pi}\dfrac{1}{1+x^2} \ \text{and} \ \text{ cdf :} \ F_C(u)=\frac{1}{2}+\frac{1}{\pi} atan{u} \ \ \ (1)$$
thus $X=k ln(C)$ with $C$ naving a Cauchy distribution. Thus the cdf of $X$ is
$$F_X(x)=P(X<x)=P(C<exp(x/k))=F_C(exp(x/k)) \ \ (2)$$
It remains to plug (1) into (2), then derive (2), in order to get its pdf.