Q:

What does intercept actually mean in a simple linear model?

Given y = a + b*x, does "a" represent simply an intercept or mean y?
For y = a + b*ln(x), is there a need to interpret the intercept "a?"

A:

In a model like
$$
E[y] = a + b*x
$$
$a$ represents the mean value of $y$ for all observations where $x = 0$. Whether or not this is meaningful will depend on whether $x=0$ is meaningful in your situation.
Similarly, in
$$
E[y] = a + b*ln(x)
$$
$a$ represents the mean value of $y$ for all observations where $ln(x)=0$, i.e. $x=1$. Again, if this is a valid situation, then you can usefully interpret $a$.

