STEM 698
The homogeneity property of power functions
For a power function f ( x)  cx k , if x is increased by a factor of m, then f is increased by a
factor of m k .
Example.
The life expectancy of certain types of stars (“main sequence stars”) such as our Sun depends on
the mass of the star. The relation is a power law: L(m)  cm2.5 where c is a constant.
a. Does a more massive star have a longer or shorter life expectancy than a less massive
star?
A more massive star has a shorter life expectancy. The negative exponent implies that the
function is decreasing.
b. The main sequence star Spica has a massive that is 7.3 times the mass of the sun. How
does the life expectancy of Spica compare to the mass of the sun?
7.32.5  0.0069 times as long.
c. Suppose a one main sequence star F4IV is half the mass of another star N9II. Fill in the
blank:
The life expectancy of F4IV is 0.52.5  5.66 times the life expectancy of N9II.
d. Suppose one star A has a life expectancy that is 10 times shorter than another star B.
How do their masses compare?
Here 10  m 2.5 . Raising both sides of the equation to the 
1
power, we get
2.5
m  0.398 . So star B has a mass that is about 0.4 times as much Star A. Another way
expressing the answer is that Star A has a mass that is about 2.5 times the mass of Star B.