\( x ^ { 2 } \log _ { 343 } ( 5 - x ) \leq \log _ { 7 } ( x ^ { 2 } - 10 x + 25 ) \)

To solve the inequality \( x^{2} \log_{343} (5 - x) \leq \log_{7} (x^{2} - 10x + 25) \), we can start by simplifying both logarithmic expressions. The expression \( \log_{343} (5 - x) \) can be rewritten using the change of base formula as \( \frac{\log_{7} (5 - x)}{3} \) because \( 343 = 7^3 \). The term \( x^{2} - 10x + 25 \) simplifies to \( (x - 5)^{2} \), which is always non-negative and equals to zero when \( x = 5 \). Now, we rewrite the inequality: \[ \frac{x^{2} \log_{7} (5 - x)}{3} \leq \log_{7} ((x - 5)^{2}) \] Multiplying all terms by 3 to eliminate the fraction gives us: \[ x^{2} \log_{7} (5 - x) \leq 3 \log_{7} ((x - 5)^{2}) \] This leads us to analyze where the logarithmic expressions are defined and how they behave. We know that \( 5 - x > 0 \) means \( x < 5 \), and \( x^{2} - 10x + 25 = 0 \) shows us that the left-hand side equals zero at \( x = 5 \) and is positive for \( x < 5 \). Now, to solve the inequality, it's essential to check values in the interval \( (-\infty, 5) \) and examine both sides. The logarithmic functions will change based on whether their arguments are positive. The inequality simplifies significantly as you explore intervals and test critical points within \( (-\infty, 5) \). In summary, the inequality will depend heavily on the properties of logarithms and the behavior of the functions involved. Careful analysis of intervals and critical points will lead to the solution. Happy solving!