Effect of quadratic term when variable's range is negative I am running a Linear Model where I want to include a quadratic term.
The dependent and the explanatory variables are all in logarithmic terms.
Further, due to Log transformation the range of X is completely negative, from -8 to -4 
I thus run:
log(y)= b0 + b1*log(x) + b2*log(X)^2 + e

... I obtain that both b1 and b2 are negative. 
Does the total effect of X increase and then decrease after a certain value of X?
Addition:
I am aware of the fact that the interpretation of the coefficient is not trivial with log variables squared. But I am only interested in signs.
Moreover, the signs with the linear variables and the quadratic terms are:
b1 >0 and b2<0. 
 A: I think what you should mean is that $\log X$ is negative with a range from $-8$ to $-4$. If $X$ itself were negative, then its logarithm is complex and not defined for this kind of statistical model. 
The answer depends entirely on the values of the coefficients, even though it is given that both are negative. Anything can happen from decreasing monotonically over your range to increasing monotonically; a turning point may or may not appear within your range. 
We can illustrate generically in terms of some $x$, here equal to your $\log X$. 

Hence there is no substitute for plotting using your estimated coefficients. 
While I was writing this, you edited so that you now first say that both coefficients are negative, and then later say that one is positive and the other is negative. That needs clarifying! Whichever it is, the principle is the same: just the plot the linear and quadratic terms to see how they behave for your observed range. (Thinking about how they would behave for rather smaller and larger values is usually a good idea too.) 
A: Although you are estimating a log-log model it is still a linear model, so the question comes down to how a quadratic function behaves.
The intercept has no bearing on the shape, so we can just consider:
$$ f(x) = ax + bx^2$$
It is clear from inspection that if $b$ is negative then the shape is a parabola that increases and then decreases, which answers the question that you posed ("Does the total effect of X increase and then decrease after a certain value of X?") and this is not affected at all by the log transformations.
Differentiating and setting to zero we find:
$$ a +2bx = 0$$
Hence, $x= -a/2b$ is the turning point (the certain value) that you mentioned, though whether this occurs within the range of your data depends on the values of $a$ and $b$.
