How much brighter will a star of 1st magnitude appear compared with one of 9th magnitude?

A star of 1st magnitude appears approximately 100 times brighter than a star of 9th magnitude. The brightness scale is logarithmic, with each whole number change in magnitude corresponding to a brightness difference of about 2.5 times. Therefore, the difference between 1st and 9th magnitude is 8 steps, resulting in a brightness factor of about (2.5^8), which equals roughly 3900 times, indicating that the 1st magnitude star is significantly brighter.