How can I prove that this has only one solution for 0< x <2 when f(x)=2?

$$f(x)= \log\_{k} (6x-3x^2)$$

When I try to solve this equation I get $$k^2=-3x(x-2)$$ 

But then I'm stuck as it feels like it leads to an infinity of results.

I sense that calculus should be involved but I don't know how or why.

Thanks.