5. Let the function $f(x)=\lg \left(a^{x}-b^{x}+9\right)(a>1$, $b>0)$, the domain of the function $f(x)$ is $[1,+\infty)$, and the range of the function $f(x)$ is a subset of $[1,+\infty)$. Then the relationship between the real numbers $a$ and $b$ is $\qquad$ .
$\begin{array}{l}\text { 5. } a \geqslant 1+b . \\ \text { When } x=1 \text {, } f(1)=\lg (a-b+9) \geqslant 1 .\end{array}$