Ytdyklly

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?

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.

$$log(Y) = alog(X) + b$$

$$exp(log(Y)) = exp(a log(X) + b)$$

$$Y = e^bcdot X^a$$

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:

$$log(Y) = a log(X) + b + epsilon$$

$$Y = e^bcdot X^acdot exp(epsilon)$$

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.

You can take your model $log(Y)=alog(X)+b$ and calculate the total differential, you will end up with something like :
$$frac{1}YdY=afrac{1}XdX$$
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 $X$.

Reconciling the answer by @Rscrill with actual discrete data, consider

$$log(Y_t) = alog(X_t) + b,;;; log(Y_{t-1}) = alog(X_{t-1}) + b$$

$$implies log(Y_t) - log(Y_{t-1}) = aleft[log(X_t)-log(X_{t-1})right]$$

But

$$log(Y_t) - log(Y_{t-1}) = logleft(frac{Y_t}{Y_{t-1}}right) equiv logleft(frac{Y_{t-1}+Delta Y_t}{Y_{t-1}}right) = logleft(1+frac{Delta Y_t}{Y_{t-1}}right)$$

$frac{Delta Y_t}{Y_{t-1}}$ 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

$$logleft(1+frac{Delta Y_t}{Y_{t-1}}right) approx frac{Delta Y_t}{Y_{t-1}}=g_{Y_{t}}$$

Therefore we get

$$g_{Y_{t}}approx ag_{X_{t}}$$

which validates in empirical studies the theoretical treatment of @Rscrill.

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 see a linear relationship.
Example I took from Google Images: