10 Given the function $f(x)=\left\{\begin{array}{ll}(3-m) x-m, & x<1, \\ \log _{m} x, & x \geqslant 1\end{array}\right.$ is monotonically increasing on the real number set $\mathbf{R}$, then the range of the real number $m$ is $\qquad$
(10) From the given, we have $\left\{\begin{array}{l}m>1, \\ 3-m>0, \\ \log _{m} 1 \geqslant(3-m)-m .\end{array}\right.$ Therefore, the solution is $\frac{3}{2} \leqslant m<3$. Hence, the answer is $\frac{3}{2} \leqslant m<3$.