Tolpygo A.K.

It is known that the number $a$ is positive, and the inequality $10<a^{x}<100$ has exactly five solutions in natural numbers.

How many such solutions can the inequality $100<a^{x}<1000$ have?
Let $a=10^{b}$. Then the inequality $10<a^{x}<100$ is equivalent to $1<b x<2$, and $100<a^{x}<1000$ is equivalent to $2<b x<3$. We have arrived at a problem analogous to $\underline{6838}$. Examples for 4, 5 or 6 solutions are obtained respectively when $b=5 / 23, 1 / 6, 5 / 27$.

## Answer

4, 5 or 6 solutions.