Answer :

Let's solve and understand the expression:

[tex]\log_5 3 \times \log_2 \left(8a^{-\frac{2}{3}} : a^{\frac{2}{3}} \right)[/tex]

For clarity, let's focus on each part step-by-step.

  1. Understanding the Expression: The given expression involves logarithms and a division inside the logarithm. We need to explore it carefully.

  2. Simplifying the Division: The expression inside the second logarithm can be rewritten as:

    [tex]8a^{-\frac{2}{3}} : a^{\frac{2}{3}} = \frac{8a^{-\frac{2}{3}}}{a^{\frac{2}{3}}}[/tex]

    This simplifies to:

    [tex]8 \times a^{-\frac{2}{3} - \frac{2}{3}} = 8 \times a^{-\frac{4}{3}}[/tex]

  3. Using Logarithm Properties: Now, consider the property:

    [tex]\log_b (MN) = \log_b M + \log_b N[/tex]

    Applying it here:

    [tex]\log_2 (8 \times a^{-\frac{4}{3}}) = \log_2 8 + \log_2 a^{-\frac{4}{3}}[/tex]

  4. Calculating Each Logarithm:

    • [tex]\log_2 8 = 3[/tex] because [tex]2^3 = 8[/tex].
    • [tex]\log_2 a^{-\frac{4}{3}} = -\frac{4}{3} \log_2 a[/tex] using the power rule: [tex]\log_b x^n = n \log_b x[/tex].

    Therefore, the expression becomes:

    [tex]3 - \frac{4}{3} \log_2 a[/tex]

  5. Combining the Expressions: Now substitute back:

    [tex]\log_5 3 \times \left(3 - \frac{4}{3} \log_2 a \right)[/tex]

    This is as simplified as the expression can get without specific values for [tex]a[/tex].

  6. Conclusion: The expression simplifies the given logs and division problem using fundamental logarithm properties frequently discussed at a high school level.

If further simplification or evaluation is needed, the specific value of [tex]a[/tex] would be required, or additional context provided by the question.