The feasible solutions are $\left(x > 0\right) \cap \left(x \left({x}^{2} - 8 x + 25\right) > 0\right)$ but for $x > 0$ we have ${x}^{2} - 8 x + 25 > 0$ with minimum at $x = 4$ with value ${4}^{2} - 8 \cdot 4 + 25 = 9$
Now $\log \left({x}^{3} - 8 {x}^{2} + 25 x\right) - \log \left(x\right) = \log \left({x}^{2} - 8 x + 25\right) = 1$ or
${x}^{2} - 8 x + 25 = e$ but as we saw previously ${x}^{2} - 8 x + 25 \ge 9$ so no real solutions for this equation.