Changing the way the curve moves for y=1/x^c with different "c" values

Question

I am working on a synthesizer application which utilizes the equation y=1/x^c for setting several parameters.

This is what the curve looks like as c moves linearly between 0 and 10:

y=1/x^c animation

As you can see, at a given x-value, like x=2, the y-axis motion has a "bounce" to it, and it does not move linearly with the linear changes in c.

I am wondering what could be used in place of just plain "c" so that, for example, at x=2, the y-axis motion follows a linear path with linear adjustment in the coefficient.

Ideally I'd like to have a place in the equation where I can specify what x value I want the linear motion to occur at. ie. Whether the coefficient should be linearly proportionate to the y-axis motion at x=2, or x=5, etc.

So in other words, I'm looking for a way to express 1/x^c so that linear changes in "c" cause proportionately linear y-axis changes at a specified x-value.

How can this be done?

Thanks.