18. Find the number of real roots of the equation $\log _{10}^{2} x-\left\lfloor\log _{10} x\right\rfloor-2=0$.
(Note: $\lfloor x\rfloor$ denotes the greatest integer not exceeding $x$.)
18. Answer: 3
Solution. Let $u=\left\lfloor\log _{10} x\right\rfloor$ and $r=\log _{10} x-u$. So $0 \leq r0$, we have $u \leq 2$. But $u+2$ is not a complete square. So $u \leq 1$. As $u+2 \geq 0$ and not a complete square, we have $u \geq 0$. Hence $u \in\{0,1\}$.
If $u=0$, then $r=\sqrt{2}-0>1$, not suitable.
If $u=1$, then $r=\sqrt{3}-1$. So $\log _{10} x=\sqrt{3}$ and $x=10^{\sqrt{3}}$.
Hence the answer is 3 .