The equation represents light cones of Schwarzschild geometry so we put ds^2 = 0 in Finkelstein coordinates for light cones
We have 2 conditions r> 2M and r<2M I am not able to solve this equation. I think I am lacking the knowledge of properties or behavior of log function. Can someone please explain me that.
The solution for outgoing is t = r +constant Where t = v-r-2M log |r/2M -1|
Here we are talking about Eddington-Finkelstein metric. Let me make you a foreword about.
We start from the Schwartzschild metric based on a spherical coordinate system:
$$ds^2 = \left(1 - \frac{2GM}{r}\right)dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2 - r^2d\Omega^2$$
where $d\Omega^2 = d\theta^2 + \sin^2(\theta)d \phi^2$
I assume you know the meaning of the variables.
Now, if we calculate the evolution for a radial geodetic ($d\Omega^2 = 0)$ null ($ds^2 = 0$) we get
$$0 = \left(1 - \frac{2GM}{r}\right)dt^2 + \left(1 - \frac{2GM}{r}\right)^{-1} dr^2$$
From which after a little easy arrangement we obtain
$$dt^2 = \frac{dr^2}{\left(1 - \frac{2GM}{r}\right)^2}$$
and thence
$$\pm dt = \frac{dr}{\left(1 - \frac{2GM}{r}\right)} = dr + \frac{2GM}{r - 2GM} dr$$
Now we integrate:
$$\pm t = \int \left(1 + \frac{2GM}{2r - 2GM}\right) dr = r + 2GM\ln\left|\frac{r}{2GM} - 1\right| + \text{constant}$$
That is: a ray of light reaches a distance $r$ from the black hole equals to $2GM$ which is the Schwarzschild radius, in an infinite amount of time (so it never does reach it).
So for an observer that is going to reach the black hole events horizon, his temporal coordinate becomes infinite (a singularity), and this is why no information can be transmitted towards the outside by an observer who is crossin the event horizon.
Given this result, Regge and Wheeler defined a new coordinate:
$$r^* = r + 2GM\ln\left|\frac{r}{2GM} - 1\right|$$
renamed "turtoise coordinate" with a strong reference to Achilles and the tortoise paradox.
Now to eliminate the singularity the ideas is to transform into straight lines the lines who enter into the horizon, substituting $t$ with a tortoise coordinate:
$$\bar{t} = t + 2GM\ln\left|\frac{r}{2GM} - 1\right|$$
In this case indeed the previous integral becomes, for $-t$,
$$\bar{t} = -r + \text{constant}$$
and the distance decreases when $\bar{t}$ increases (inwards geodetics).
So substituting $\bar{t}$ in the initial metrics, we get Eddington-Finkelstein inwards coordinates
$$ds^2 = \left(1 - \frac{2GM}{r}\right)d\bar{t}^2 -\frac{4GM}{r}d\bar{t}dr - \left(1 - \frac{2GM}{r}\right)dr^2 - r^2d\Omega^2$$
Defining now new coordinates such that $$v = \bar{t} + r$$
(know as advanced time), we can simplify the metrics:
$$ds^2 = \left(1 - \frac{2GM}{r}\right) dv^2 -2 dv dr - r^2d\Omega^2$$
and we get as a solution
$$v = 2r^* + \text{constant}$$
from which
$$\bar{t} = r - 4\ln\left|\frac{r}{2GM}-1\right|$$
This tells us that while the time increases, the distance from the centre of the black hole decreases. This does indeed describes the time evolution of an object in a presence of a black hole.
In a similar way we can get the outwards Eddington-Finkelstein coordinates, which would describe a mythical object known as white hole.
So, I suspect there is much more work to do than knowing some logarithm properties (which indeed were not even used here).