1. **State the problem:** Simplify the expression $$\log_a(8) + \frac{4}{5} \log_a(14) - \frac{4}{5} \log_a(7) + \log_a\left(\left(\frac{1}{2}\right)^{\frac{9}{5}}\right)$$ using properties of logarithms. 2. **Recall logarithm properties:** - Power rule: $$\log_a(x^r) = r \log_a(x)$$ - Product rule: $$\log_a(x) + \log_a(y) = \log_a(xy)$$ - Quotient rule: $$\log_a(x) - \log_a(y) = \log_a\left(\frac{x}{y}\right)$$ 3. **Apply the power rule to terms with coefficients:** $$\frac{4}{5} \log_a(14) = \log_a(14^{\frac{4}{5}})$$ $$\frac{4}{5} \log_a(7) = \log_a(7^{\frac{4}{5}})$$ 4. **Rewrite the expression:** $$\log_a(8) + \log_a(14^{\frac{4}{5}}) - \log_a(7^{\frac{4}{5}}) + \log_a\left(\left(\frac{1}{2}\right)^{\frac{9}{5}}\right)$$ 5. **Combine the subtraction into a quotient using the quotient rule:** $$\log_a(8) + \log_a\left(\frac{14^{\frac{4}{5}}}{7^{\frac{4}{5}}}\right) + \log_a\left(\left(\frac{1}{2}\right)^{\frac{9}{5}}\right)$$ 6. **Simplify the fraction inside the log:** $$\frac{14^{\frac{4}{5}}}{7^{\frac{4}{5}}} = \left(\frac{14}{7}\right)^{\frac{4}{5}} = 2^{\frac{4}{5}}$$ 7. **Rewrite the expression:** $$\log_a(8) + \log_a\left(2^{\frac{4}{5}}\right) + \log_a\left(2^{-\frac{9}{5}}\right)$$ 8. **Combine the last two terms using the product rule:** $$\log_a\left(2^{\frac{4}{5}} \times 2^{-\frac{9}{5}}\right) = \log_a\left(2^{\frac{4}{5} - \frac{9}{5}}\right) = \log_a\left(2^{-1}\right)$$ 9. **Rewrite the entire expression:** $$\log_a(8) + \log_a\left(2^{-1}\right) = \log_a\left(8 \times 2^{-1}\right) = \log_a(4)$$ **Final answer:** $$\boxed{\log_a(4)}$$