How much has Mars dimmed two weeks after its brightest showing on August 25th?

On the 25th, Mars was closest to Earth so they were aligned, with Mars at point A. Two weeks later Mars was at B and Earth had moved to the position shown. From references on planetary motion (e.g. the back of your text), the orbital period of Mars is 1.88 years (686.973 days), and that of Earth is 1 year (365.242 days). From this it follows that in two weeks Mars moves through an angle of  14÷686.973×360 = 7.33653°. During the same interval Earth moves through an angle of 14÷365.242×360 =  13.7991°. This makes the angle between Earth and Mars at the later time to be 13.8 - 7.34, or 6.46254°.

Also from the references, Earth orbits at a distance of 149.60×10^6km while Mars orbits at 227.92×10^6 km.

From the figure, at closest approach the Mars-Earth distance is r_m - r_e = 78.320×10^6km and at the 2-week point the distance is (from the law of cosines) FormBox[RowBox[{RowBox[{(, RowBox[{r_m^2, +, r_e^2, -, RowBox[{2, r_m, r_e, RowBox[{Cos, (, RowBox[{6.46, °}], )}]}]}], )}], ^, (1/2)}], TraditionalForm]= 81.0392×10^6km.

Light falls off as the square of the distance, so the reduction in brightness after two weeks should be FormBox[RowBox[{RowBox[{RowBox[{(, RowBox[{78.32, ×, 10^6}], )}], ^, 2}], }], TraditionalForm]÷ FormBox[RowBox[{RowBox[{(, RowBox[{81.0392, ×, 10^6}], )}], ^, 2}], TraditionalForm], or about 6.6% of its brightness. The calculations are shown below.
    

[Graphics:HTMLFiles/marscalc_10.gif]

RowBox[{, RowBox[{RowBox[{RowBox[{r_e, =, RowBox[{149.6, ×, 10^6}]}], ;, , RowBox[{r_ ... /r_f^2, , di = i_i - i_f, , i_f/i_i, , (i_i - i_f) * 100/i_i, }]}]

7.832*10^7

13.7991

7.33653

6.46254

8.10392*10^7

2.71918*10^6

1.63025*10^-16

1.52268*10^-16

1.07567*10^-17

0.934018

6.59818

Created by Mathematica  (September 9, 2003)