Illustrate applications of the logarithm.
Stars have an apparent magnitude m, which is the brightness of light reaching Earth. They also have an absolute magnitude M, which is the intrinsic brightness and does not depend on the distance from Earth. The difference S = m − M is the spectroscopic parallax. Spectroscopic parallax is related to the distance D from Earth, in parsecs, by
S = 5 log D − 5.
(a) Use the laws of logarithms to determine what happens to S if D is doubled. (Round your answer to two decimal places.)
If D is doubled S increases by ______ (I got 1.51)
(b) Solve the equation above for D to express distance as a function of spectroscopic parallax.
S = 5 log D − 5
a. If D is doubled
S = 5 log (2D) - 5 =
5 [ log 2 + log D] - 5 =
5 log2 +[ 5logD - 5 ] =
5log2 + S
5log2 ≈ 1.51
So "S" increases by ≈ 1.51
b. S = 5 log D − 5 add 5 to both sides
S + 5 = 5logD divide both sides by 5
[S + 5] / 5 = log D exponentially, we have that
D = 10 ( [S + 5] / 5 ) = 10 s/5 + 1