How many roots?#2

Since $\log _{ 100 }{ x } =\sin { x } \le 1\Longrightarrow x\le 100$ .

Let's study the graphs of $y=\log _{ 100 }{ x }$ and $y=\sin { x }$ in (0,100] below

• In the first interval $\left( 0,2\pi \right)$ , the equation has only 1 positive root.

• In the next 14 intervals $\left( k2\pi ,\left( k+1 \right) 2\pi \right)$ , $k=1,2,3,...,14$ we notice that

  If $x=\frac { \pi }{ 2 } +k2\pi \Longrightarrow \sin { x } =1>\log _{ 100 }{ x } ,$

  If $x=\pi +k2\pi \Longrightarrow \sin { x } =0<\log _{ 100 }{ x } .$

Hence, each interval gives 2 intersections of 2 graphs, i.e. the equation has 2 roots each interval.

• In the last interval $\left( 15.2\pi ,100 \right)$ , the length of the interval $100-15.2\pi >5>\pi$ , so that interval "contains" the part above $x-$ axis $\Longrightarrow$ we also have 2 roots in the last interval.

Finally, the number of roots of the provided equation are $1+14\times 2+2=\boxed { 31 } .$

Its graph is pretty interesting.