Logarithm Expression 4B9Ad6

1. **State the problem:** Express the expression $$\frac{1}{2} \lg 25 - 3$$ as a single logarithm. 2. **Recall logarithm properties:** - Power rule: $$a \log b = \log b^a$$ - Difference rule: $$\log a - \log b = \log \frac{a}{b}$$ - Logarithm of a number: $$\log 10^k = k$$ if base is 10 (common logarithm). 3. **Rewrite the terms:** $$\frac{1}{2} \lg 25 = \lg 25^{\frac{1}{2}} = \lg \sqrt{25} = \lg 5$$ 4. **Rewrite the constant 3 as a logarithm:** Since $$\lg 10 = 1$$, then $$3 = \lg 10^3 = \lg 1000$$ 5. **Combine the logarithms using the difference rule:** $$\lg 5 - \lg 1000 = \lg \frac{5}{1000} = \lg \frac{1}{200}$$ 6. **Final answer:** $$\boxed{\lg \frac{1}{200}}$$