Example 1

Arcturus, the brightest star in the northern summer constellation Bootes the Herdsman, has a g-r color of 1.06. How many times more red light does Arcturus emit, as compared to its green light?

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.