# Weighted sums of power curve

I am not really sure how to even phrase this question, but here it goes:

I'm looking at a distribution that follows a power curve (I think).. it looks something like this:

f(0):  100000
f(1):   10000
f(10):   1000
f(100):   100
f(1000):   10
f(10000):   1
f(100000): .1

1. Is this a power curve?

2. Am I to take it that for all power curves, that $x \cdot f(x)$, and also $\log(x) + \log(f(x))$ is constant, or is this a special case?

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Aside from $f(0)$ they follow $f(x)=\frac {10000}x$ perfectly. This is a power curve with exponent $-1$
For 2, $x \cdot f(x)$ is only constant if the exponent is $-1$. $\log x + \log f(x) = \log (x \cdot f(x))$ so this will be constant again when the exponent is $-1$