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 Homework Statement
 A UBV photometric (UBV Johnson’s) observation of a star gives U = 8.15, B = 8.50, and V = 8.14. Based on the spectral class, one gets the intrinsic color (U – B)o = 0.45. Determine the intrinsic magnitudes M_U, M_B, and M_V of the star (take, for the typical interstellar matters, the ratio of total to selective extinction R = 3.2). The star is known to have radius of 2.3 solar radii, absolute bolometric magnitude of 0.25, and bolometric correction (BC) of 0.15.
 Homework Equations

B  V = (B  V)o + E(B  V)
U  B = (U  B)o + E(U  B)
R = Av / E(B  V) = 3.2
m  M = 5log(r/10) + A
E(U  B) = 0.72*E(B  V)
I first determined ##E_{U  B}## by using the second equation listed above: $$U  B = (U  B)_0 + E_{U  B}$$ $$8.15  8.50 = 0.45 + E_{U  B}$$ Then since the ratio to the BV color excess is known, I solved for that and obtained ##E_{B  V} = 0.14##. Using the ratio of total to selective extinction and the fact that ##E_{BV} = A_B  A_V##, we get ##A_V = 0.44## and ##A_B = 0.58##. (So far this is in line with the given solution.)
From here, the solution given for this problem takes the following step: it uses ##V = M_V + A_V##, giving ##8.14 = M_V + 0.44## and solving for ##M_V##, ##M_B##, and ##M_U## from there, obtaining ##M_V = 7.69##, ##M_B = 7.91##, and ##M_U = 7.46##.
I don't understand why this is a valid method. I've looked in my textbook and it corroborates what I assumed to be correct  that the equation for V is ##V = M_V + 5log\frac{r}{10} + A_V## to account for the "spreading out" of the light in addition to the extinction. So is the ##5log\frac{r}{10}## automatically accounted for in ##A_V## in this case? If not, what am I missing? Thanks to anyone who can offer help.
There is some extra information in the homework statement that was used for further parts of this question; I don't believe it's relevant to this part, but I left it there in case it has some implication here. One other possibility is that I am misunderstanding the term "intrinsic magnitude" in the question statement, which I took to mean "absolute magnitude." Is there a difference here?
From here, the solution given for this problem takes the following step: it uses ##V = M_V + A_V##, giving ##8.14 = M_V + 0.44## and solving for ##M_V##, ##M_B##, and ##M_U## from there, obtaining ##M_V = 7.69##, ##M_B = 7.91##, and ##M_U = 7.46##.
I don't understand why this is a valid method. I've looked in my textbook and it corroborates what I assumed to be correct  that the equation for V is ##V = M_V + 5log\frac{r}{10} + A_V## to account for the "spreading out" of the light in addition to the extinction. So is the ##5log\frac{r}{10}## automatically accounted for in ##A_V## in this case? If not, what am I missing? Thanks to anyone who can offer help.
There is some extra information in the homework statement that was used for further parts of this question; I don't believe it's relevant to this part, but I left it there in case it has some implication here. One other possibility is that I am misunderstanding the term "intrinsic magnitude" in the question statement, which I took to mean "absolute magnitude." Is there a difference here?