Logarithm Expression 6645Fb

1. **State the problem:** We are given that $\log_a x = 2$, $\log_a y = 5$, and $\log_a z = -3$. We need to find the value of $\log_a\left(\frac{xy^3}{z^2}\right)$. 2. **Recall logarithm properties:** - $\log_a (MN) = \log_a M + \log_a N$ - $\log_a \left(\frac{M}{N}\right) = \log_a M - \log_a N$ - $\log_a (M^k) = k \log_a M$ 3. **Apply the properties:** $$\log_a\left(\frac{xy^3}{z^2}\right) = \log_a (x) + \log_a (y^3) - \log_a (z^2)$$ 4. **Use the power rule:** $$= \log_a x + 3 \log_a y - 2 \log_a z$$ 5. **Substitute the given values:** $$= 2 + 3 \times 5 - 2 \times (-3)$$ 6. **Calculate step-by-step:** $$= 2 + 15 + 6$$ 7. **Sum the values:** $$= 23$$ **Final answer:** $\boxed{23}$