You know that Arcturus’s g-r color is 1.06. By the definition of color:
g-r = Magg - Magr.
And by the definition of magnitude,
g-r = -log2.51 (Fgreen) + log2.51 (Fred) = 1.06.
Switch the order of the logarithms to get log2.51 (Fred) – log2.51 (Fgreen) = 1.06.
Using the rule of logarithms that log (x) – log (y) = log (x/y), you know that:
log2.51 (Fred / Fgreen) = 1.06.
Invert the log by raising 2.51 to the power of both sides of the equation; that is,
2.51 ^ [log2.51 (Fred / Fgreen)] = 2.511.06
The log is cancelled on the left side of the equation, meaning:
Fred / Fgreen = 2.511.06.
To solve for Fred, get Fred alone on the left side of the equation:
Fred = (2.511.06)Fgreen.
Use a calculator to find that the coefficient is about 2.65.
Therefore, Arcturus gives off 2.65 times as much red light as green light.