The magnitude M of a star is a measure of how bright a star appears to the human eye. It is defined by M = -2.5 log() where B is the actual brightness of the star and Bo is a constant. (a) Expand the right-hand side of the equation. (b) Use part (a) to show that the brighter a star, the less its magnitude. Suppose B1 and B2 are the brightness of two stars such that B1 < B2, and let M1 and M2 be their respective magnitudes. Since log is an increasing function, we have log(B1) ? log(B2) log(B1) – log(Bo) ? log(B2) – log(Bo) log ? log -2.5 log 2) ? -2.5 log M1 ?v M2 Thus, the brighter star has less magnitude. (c) Betelgeuse is about 100 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less bright than Albiero. Let B, be the brightness the star and M, be magnitude. Then 100B1 the brightness Betelgeuse, and magnitude is M = -2.5 log = -2.5 log B1 + log Во (B1 + lol = -2.5 Bo B1 2.5 log = + M1.
The magnitude M of a star is a measure of how bright a star appears to the human eye. It is defined by M = -2.5 log() where B is the actual brightness of the star and Bo is a constant. (a) Expand the right-hand side of the equation. (b) Use part (a) to show that the brighter a star, the less its magnitude. Suppose B1 and B2 are the brightness of two stars such that B1 < B2, and let M1 and M2 be their respective magnitudes. Since log is an increasing function, we have log(B1) ? log(B2) log(B1) – log(Bo) ? log(B2) – log(Bo) log ? log -2.5 log 2) ? -2.5 log M1 ?v M2 Thus, the brighter star has less magnitude. (c) Betelgeuse is about 100 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less bright than Albiero. Let B, be the brightness the star and M, be magnitude. Then 100B1 the brightness Betelgeuse, and magnitude is M = -2.5 log = -2.5 log B1 + log Во (B1 + lol = -2.5 Bo B1 2.5 log = + M1.
Chapter6: Exponential And Logarithmic Functions
Section6.3: Logarithmic Functions
Problem 66SE: The intensitylevels I of two earthquakes measured ona seismograph can be compared by the formula...
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Transcribed Image Text:The magnitude M of a star is a measure of how bright a star appears to the human eye. It is defined by
M = -2.5 log()
Bo
where B is the actual brightness of the star and Bo is a constant.
(a) Expand the right-hand side of the equation.
(b) Use part (a) to show that the brighter a star, the less its magnitude.
Suppose B1 and B2 are the brightness of two stars such that B1 < B2, and let M1 and M2 be their respective magnitudes. Since log is an increasing function, we have
log(B1)
? v log(B2)
log(B1) – log(Bo) [?v log(B2) – log(Bo)
B1
log
Во
?v log
B1
-2.5 log (
? v -2.5 log
Во
M1 ? v M2
Thus, the brighter star has less magnitude.
(c) Betelgeuse is about 100 times brighter than Albiero. Use part (a) to show that Betelgeuse is 5 magnitudes less bright than Albiero.
Let B1 be the brightness of the star Albiero and M1 be its magnitude. Then 100B1 is the brightness of Betelgeuse, and its magnitude is
М— —2.5 log
B1
+ log
Bo
= -2.5 log|
B1
+ log
Во
= -2.5
B1
2.5 log
Во
+ M1.
I| ||
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