To solve the logarithmic equation \( y = \log_b x \) for the given point \( \left( \frac{1}{6}, -1 \right) \), we can substitute the values into the equation.
We have:
\[ y = -1 \quad \text{and} \quad x = \frac{1}{6} \]
Substituting these values into the logarithmic equation gives us:
\[ -1 = \log_b \left( \frac{1}{6} \right) \]
By the definition of logarithms, we can rewrite the equation:
\[ b^{-1} = \frac{1}{6} \]
This means:
\[ \frac{1}{b} = \frac{1}{6} \]
To find \( b \), we can take the reciprocal of both sides:
\[ b = 6 \]
Thus, the value of \( b \) is
\[ \boxed{6} \]