Logarithmic Differentiation When do we use :  $ \ln(ab) = \ln a + \ln b $ and when do we use : $ \ln |y| = \ln |f_1(x)| + \ln |f_2(x)| + \cdots + \ln |f_n(x)| $ ? 
It is stated that we use the second form of log differentiation when, $y = f_1(x)f_2(x)\cdots f_n(x)$ is a product of nonzero functions. 
What do they mean by product of nonzero functions?
Example given for second differentiation: 
Find $dy/dx$ if $$ y =\frac{x \cos (x)}{\sqrt{\csc (x)}}. $$
 A: They mean that
$$y(x) = f_1(x) \times f_2(x) \times \ldots \times f_n(x)$$
where none of the terms on the right-hand side are zero.
Then you have
$$
\ln |y| = \ln \left| \prod_{k=1}^n f_k(x) \right|
        = \ln \left( \prod_{k=1}^n \left| f_k(x) \right| \right)
        = \sum_{k=1}^n \ln \left| f_k(x) \right|
$$
Rewriting equivalently in more usual notation:
$$\begin{split}
\ln |y|
 &= \ln \left| f_1(x) \times f_2(x) \times \ldots \times f_n(x) \right| \\
 &= \ln \left( \left| f_1(x) \right| \times
               \left| f_2(x) \right| \times \ldots \times
               \left| f_n(x) \right| \right) \\
 &= \ln \left| f_1(x) \right| + \ln \left| f_2(x) \right|
                              + \ldots + \ln \left| f_n(x) \right|.
\end{split}$$
A: A function $f(x)$ is said to be non-zero if for any $x \in \mathbb R$ $f(x) \ne 0$ ,i.e does not have a value $0$. 
This is necessary because the function $\log_a(x)$ is undefined at $x=0$.
After that it is simply an application of the law, 
$$\log_k(ab) = \log_k(a) +\log_k(b)$$
For proof,
$$\text{Let }k^x = a, k^y = b$$
$$\implies \log_k(a)=x, \log_k(b)=y$$
$$ab=k^xk^y=k^{x+y}$$
$$\implies \log_k(ab) = x+y$$
