Solution:

The answer is 4. The equation can be written in the form $2^{5x}=2^{6[x]}$, which is equivalent to $5x=6[x]$. Since $x<[x]+1$, we have that $6[x]=5x<5[x]+5$, from which $[x]<5$. Since for every $x>0$ the integer part $[x]$ is an integer $\geq 0$, only the possibilities $[x]=0,1,2,3,4$ remain. Substituting these values into the relation already found $5x=6[x]$, we obtain that the corresponding values of $x$ are, respectively, $0$, $6/5$, $12/5$, $18/5$, $24/5$. The solution $x=0$ must however be discarded, since the text required $x$ positive. One easily verifies instead that the other four values of $x$ actually solve the proposed equation, which therefore has four positive real solutions.