Step 1: Convert the decimal to a fraction and simplify. $$0.6125 = \frac{6125}{10000}$$ Divide both numerator and denominator by common factors. $$ \frac{6125}{10000} = \frac{1225}{2000} = \frac{245}{400} = \frac{49}{80} $$ So, we need to evaluate $\log \left(\frac{49}{80}\right)$. Step 2: Express the fraction in terms of prime factors. $$49 = 7^2$$ $$80 = 8 \times 10 = 2^3 \times (2 \times 5) = 2^4 \times 5$$ Therefore, $$\log 0.6125 = \log \left(\frac{7^2}{2^4 \times 5}\right)$$ Step 3: Apply logarithm properties. Using the properties $\log \left(\frac{A}{B}\right) = \log A - \log B$, $\log (A \times B) = \log A + \log B$, and $\log (A^n) = n \log A$: $$\log \left(\frac{7^2}{2^4 \times 5}\right) = \log (7^2) - \log (2^4 \times 5)$$ $$= 2 \log 7 - (\log (2^4) + \log 5)$$ $$= 2 \log 7 - 4 \log 2 - \log 5$$ Step 4: Determine the value of $\log 2$. We are given $\log 5 = 0.6990$. Assuming base 10 logarithms (standard when no base is specified and values like 0.6990 are given), we know $\log 10 = 1$. Since $10 = 2 \times 5$, we have: $$\log 10 = \log (2 \times 5)$$ $$1 = \log 2 + \log 5$$ $$1 = \log 2 + 0.6990$$ $$\log 2 = 1 - 0.6990$$ $$\log 2 = 0.3010$$ Step 5: Substitute the known values and calculate. Given $\log 5 = 0.6990$ and $\log 7 = 0.8451$. Substitute these values and $\log 2 = 0.3010$ into the expression from Step 3: $$\log 0.6125 = 2(0.8451) - 4(0.3010) - 0.6990$$ $$= 1.6902 - 1.2040 - 0.6990$$ $$= 0.4862 - 0.6990$$ $$= -0.2128$$ The final answer is $\boxed{-0.2128}$. 3 done, 2 left today. You're making progress.