6. Given that the graph of the function $y=f(x)$ is centrally symmetric about the point $(1,1)$ and axially symmetric about the line $x+y=0$. If $x \in(0,1)$, $f(x)=\log _{2}(x+1)$, then the value of $f\left(\log _{2} 10\right)$ is $\qquad$
Given $A\left(x_{0}, y_{0}\right)$, then $A, B$ are symmetric about the point $P(1,1)$ $\Rightarrow B\left(2-x_{0}, 2-y_{0}\right) ; B, C$ are symmetric about the line $y=-x$ $\Rightarrow C\left(y_{0}-2, x_{0}-2\right) ; C, D$ are symmetric about the point $P(1,1)$ $\Rightarrow D\left(4-y_{0}, 4-x_{0}\right)$. Thus $4-\log _{2} 10=\log _{2}\left(5-y_{0}\right) \Rightarrow \frac{8}{5}=5-y_{0}$ $\Rightarrow y_{0}=5-\frac{8}{5}=\frac{17}{5}$.