How bright a star appears can depend on how much light the star actually emits and how far away it is. The stellar magnitude scale can be adjusted to account for distance as follows:

M2-M1 = log (b1 / b2)

Here, M refers to a star's absolute magnitude, that is, how brightly it appears from a standard distance of 10 parsecs (or 32.6 light-years). The absolute brightness of Sirius is 1.4 and the absolute brightness of Betelgeuse is -8.1.

Which of these two stars is brighter, in absolute terms, and by how much?

To determine which star is brighter in absolute terms and by how much, we can compare their absolute magnitudes using the given equation:

M2 - M1 = log (b1 / b2)

Given:
Absolute brightness of Sirius (b1) = 1.4
Absolute brightness of Betelgeuse (b2) = -8.1

We can substitute these values into the equation:

M2 - M1 = log (1.4 / 10^(8.1))

To calculate this, we need to understand logarithms. The logarithm function gives us the exponent to which a base must be raised to obtain a given value. In this case, we are using a base of 10.

Using a scientific calculator or math software, we can calculate the logarithm:

M2 - M1 = log (1.4 / 10^8.1) ≈ log (1.4 / 126000000)

The result of this calculation is approximately -8.997.

Therefore, the difference in absolute magnitude between Sirius and Betelgeuse is approximately -8.997. And since absolute magnitude is a measure of brightness, we can conclude that Betelgeuse is brighter (in absolute terms) than Sirius by approximately 8.997 magnitudes.