1. Stating the problem: Simplify the equation $a=\sqrt{xy}$ after multiplying both sides by the negative logarithm to base 10. 2. Start by writing the original equation: $$a=\sqrt{xy}$$ 3. Multiply both sides by the negative logarithm to base 10, denoted as $-\log_{10}$: $$-\log_{10}(a) = -\log_{10}(\sqrt{xy})$$ 4. Use the logarithm power rule $\log_b(m^n) = n \log_b(m)$ to simplify the right side: $$-\log_{10}(a) = -\log_{10}((xy)^{1/2}) = -\left(\frac{1}{2}\log_{10}(xy)\right)$$ 5. Distribute the negative sign: $$-\log_{10}(a) = -\frac{1}{2} \log_{10}(xy)$$ 6. Alternatively, we can write: $$-\log_{10}(a) = -\frac{1}{2}(\log_{10}(x) + \log_{10}(y))$$ This is the simplified form after multiplying both sides by the negative logarithm to base 10. Final answer: $$-\log_{10}(a) = -\frac{1}{2} (\log_{10}(x) + \log_{10}(y))$$