To find \( \log 6 \) using the given information \( \log 36 \approx 1.556 \), we can use the property of logarithms that relates the logarithm of a number and its square. Specifically, we know that:
\[ \log 36 = \log(6^2) = 2 \log 6 \]
From this equation, we can isolate \( \log 6 \):
\[ \log 6 = \frac{\log 36}{2} \]
Now substituting the approximate value of \( \log 36 \):
\[ \log 6 = \frac{1.556}{2} \approx 0.778 \]
Thus, the value of \( \log 6 \) to the nearest thousandth is:
\[ \boxed{0.778} \]
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