<p>Find the sum of possible real values of <i>x</i> for which the sixth term of <br/><span class="math">\left(\frac{1}{7^{\log_7(3|x-2|-9)}} + 7^{\log_3 9|x-2|} \cdot \frac{5}{3}\right)^7</span><br/> equals 567.</p>
Step-by-Step Solution
Key Concept: Use properties of logarithms to simplify the base expressions, then apply binomial theorem to find the sixth term.
<p><strong>Solution:</strong> Using the binomial expansion, the sixth term of the expansion is given by \(T_6 = \binom{7}{5}a^2b^5\), where \(a = \frac{1}{7^{\log_7(3|x-2|-9)}}\) and \(b = 7^{\log_3 9|x-2|} \cdot \frac{5}{3}\).</p><p>Simplifying: \(a = \frac{1}{3|x-2|-9}\) and \(b = (3|x-2|)^{\log_3(5/3)}\).</p><p>After simplification and solving the resulting equation, we find the real values of <i>x</i>. The sum of all such values equals <strong>4</strong>.</p>
Correct Answer: 4