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 $\log \left(\frac{49}{80}\right)$.

$$49 = 7^2$$ $$80 = 8 \times 10 = 2^3 \times (2 \times 5) = 2^4 \times 5$$

Therefore,

$$\log 0.6125 = \log \left(\frac{7^2}{2^4 \times 5}\right)$$

Using the properties $\log \left(\frac{A}{B}\right) = \log A - \log B$, $\log (A \times B) = \log A + \log B$, and $\log (A^n) = n \log A$:

$$\log \left(\frac{7^2}{2^4 \times 5}\right) = \log (7^2) - \log (2^4 \times 5)$$ $$= 2 \log 7 - (\log (2^4) + \log 5)$$ $$= 2 \log 7 - 4 \log 2 - \log 5$$

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:

$$\log 10 = \log (2 \times 5)$$ $$1 = \log 2 + \log 5$$ $$1 = \log 2 + 0.6990$$ $$\log 2 = 1 - 0.6990$$ $$\log 2 = 0.3010$$

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 $-0.2128$.