1. **State the problem:** Simplify the expression using logarithmic and exponential properties: $$5 \log 10^{-7} - 3 \ln \sqrt{e^6}$$ 2. **Recall the properties:** - $\log a^b = b \log a$ - $\ln e^x = x$ - $\sqrt{a} = a^{1/2}$ 3. **Simplify each term:** - For $5 \log 10^{-7}$, use $\log 10 = 1$, so: $$5 \log 10^{-7} = 5 \times (-7) \log 10 = 5 \times (-7) \times 1 = -35$$ - For $3 \ln \sqrt{e^6}$, rewrite the square root: $$\sqrt{e^6} = (e^6)^{1/2} = e^{6 \times \frac{1}{2}} = e^3$$ - Then: $$3 \ln e^3 = 3 \times 3 = 9$$ 4. **Combine the terms:** $$-35 - 9 = -44$$ **Final answer:** $$\boxed{-44}$$