Function Analysis Monotony $f(x)=\frac{\ln x^2-2x-3}{\ln \frac{1}{2}}$ I would like to determine whether the function 
$$f(x)=\frac{\ln (x^2-2x-3)}{\ln (\frac{1}{2})}$$
is monotone.
I have computed
$$f''(x)=\dfrac{2\left(x^2-2x+5\right)}{\ln\left(2\right)\left(x^2-2x-3\right)^2}.$$
What should I do next?
 A: Why check the second derivative? The first derivative is enough to find out the intervals in which the function is increasing or decreasing.

A: Ok so as has been pointed out it is the first derivative you want to be investigating, but even so you have
$$
\begin{align}
\ln \left( x^2 - 2x - 3 \right) &= \ln \left[(x+1)(x-3) \right] \\
&= \ln(x+1) + \ln(x-3)
\end{align}
$$
Now if you are happy that both functions on the right-hand side are monotone then it as an easy exercise that the sum of monotone functions is again monotone. There are some questions hanging around regarding domains of definition and so on, but my point would be that don't just robotoically carry out differentiation etc. 
A: You are erroneously considering the second derivative, which does not relate to monotonicity. As exhibited in the result below, you should instead consider the sign of the first derivative.

Proposition: Let $f$ be a function (real input, real output). Suppose that $f$ has a continuous derivative, $f^{\prime}$. If $[a,b]$ is an interval on which $f^{\prime}\geq0$, $f$ is monotonically increasing on $[a,b]$.
Proof: By the fundamental theorem of calculus, we know that for any $x\in[a,b]$, $$ f(x)=f(a)+\int_{a}^{x}f^{\prime}(t)dt. $$ Then, for $y,z\in[a,b]$ with $y<z$, $$f(z)-f(y)=\int_{a}^{z}f^{\prime}(t)dt-\int_{a}^{y}f^{\prime}(t)dt=\int_{y}^{z}f^{\prime}(t)dt\geq0.$$ Moving some terms around, we get $f(z)\geq f(y)$, as desired.

Hint: Understand the above result. Compute the first derivative of your function and examine it (plotting it roughly might help).
A: The function $Ln(.)$ is always monotone. $x^2-2x-3$ is also monotone (in the related domain of main function $f$). and by an easy proof we can prove that the composition of two monotone functions led to a monotone function.
