What is the intuitive meaning of having a linear relationship between the logs of two variables? I have two variables which don't show much correlation when plotted against each other as is, but a very clear linear relationship when I plot the logs of each variable agains the other.
So I would end up with a model of the type:
$$\log(Y) = a \log(X) + b$$ , which is great mathematically but doesn't seem to have the explanatory value of a regular linear model. 
How can I interpret such a model?  
 A: Intuitively $\log$ gives us the order of magnitude of a variable, so we can view the relationship as the orders of magnitudes of the two variables are linearly related. For example, increasing the predictor by one order of magnitude may be associated with an increase of three orders of magnitude of the response. 
When plotting using a log-log plot we hope to see a linear relationship.
Using an example from this question, we can check the linear model assumptions:

A: Reconciling the answer by @Rscrill with actual discrete data, consider
$$\log(Y_t) = a\log(X_t) + b,\;\;\; \log(Y_{t-1}) = a\log(X_{t-1}) + b$$
$$\implies \log(Y_t) - \log(Y_{t-1}) = a\left[\log(X_t)-\log(X_{t-1})\right]$$
But 
$$\log(Y_t) - \log(Y_{t-1}) = \log\left(\frac{Y_t}{Y_{t-1}}\right) \equiv \log\left(\frac{Y_{t-1}+\Delta Y_t}{Y_{t-1}}\right) = \log\left(1+\frac{\Delta Y_t}{Y_{t-1}}\right)$$
$\frac{\Delta Y_t}{Y_{t-1}}$ is the percentage change of $Y$ between periods $t-1$ and $t$, or the growth rate of $Y_t$, say $g_{Y_{t}}$. When it is smaller than $0.1$, we have that an acceptable approximation is
$$\log\left(1+\frac{\Delta Y_t}{Y_{t-1}}\right) \approx \frac{\Delta Y_t}{Y_{t-1}}=g_{Y_{t}}$$
Therefore we get 
$$g_{Y_{t}}\approx ag_{X_{t}}$$
which validates in empirical studies the theoretical treatment of @Rscrill.
A: You just need to take exponential of both sides of the equation and you will get a potential relation, that may make sense for some data.
$$\log(Y) = a\log(X) + b$$
$$\exp(\log(Y)) = \exp(a \log(X) + b)$$
$$Y = e^b\cdot X^a$$
And since $e^b$ is just a parameter that can take any positive value, this model is equivalent to:
$$Y=c \cdot X^a$$
It should be noted that model expression should include the error term, and these change of variables has interesting effects on it:
$$\log(Y) = a \log(X) + b + \epsilon$$
$$Y = e^b\cdot X^a\cdot \exp(\epsilon)$$
That is, your model with a additive errors abiding to the conditions for OLS (normally distributed errors with constant variance) is equivalent to a potential model with multiplicative errors whose logaritm follows a normal distribution with constant variance.
A: A linear relationship between the logs is equivalent to a power law dependence: 
$$Y \sim X^\alpha$$
In physics such behavior means that the system is scale free or scale invariant. As an example, if $X$ is distance or time this means that the dependence on $X$ cannot be characterized by a characteristic length or time scale (as opposed to exponential decays). As a result, such a system exhibits a long-range dependence of the $Y$ on $X$.
A: You can take your model $\log(Y)=a\log(X)+b$ and calculate the total differential, you will end up with something like :
$$\frac{1}YdY=a\frac{1}XdX$$
which yields to 
$$\frac{dY}{dX}\frac{X}{Y}=a$$
Hence one simple interpretation of the coefficient $a$ will be the percent change in $Y$ for a percent change in $X$. 
This implies furthermore that the variable $Y$ growths at a constant fraction ($a$) of the growth rate of $X$.
