I'm trying to follow http://www2.mpia-hd.mpg.de/~dullemon/lectures/fluiddynamics08/chap_2_hyperbolic.pdf and am confused by equation 2.5 and 2.6. To summarize, they define $\xi = \int^{x}_{x_0} \frac{dx'}{u(x')}$ to state the solution to the advection equation $\partial_t q(x,t) + u(x) \partial_x q(x,t) = 0$
as
$q(x,t) = q(x(\xi),t) = q(x(\xi-t),t)$,
"where $x(\xi)$ is the value of x belonging to the value of $\xi$."
I'm finding that wording confusing and thus the semantics elude me. If I let u(x)=1 and $x_0$=0, the formula provides intuitive answers; q(10,1)=q(9,0). But if I try a more complicated example such as u(x)=x and $x_0$=1, I get $\xi=ln(x)$ and when I let x=e,t=1 I find q(e,1)=q(ln(e)-1,0) = q(0,0), which is before my x0.
Have I made a mistake in my example? If not, am I interpreting the result incorrectly?