Given the logarithmic equation \( y = \log_b x \), and the point \( \left( \frac{1}{6}, -1 \right) \), we can substitute \( x = \frac{1}{6} \) and \( y = -1 \) into the equation:
\[ -1 = \log_b \left( \frac{1}{6} \right) \]
By converting the logarithmic form to exponential form, we rewrite this as:
\[ b^{-1} = \frac{1}{6} \]
This implies:
\[ \frac{1}{b} = \frac{1}{6} \]
Taking the reciprocal of both sides, we find:
\[ b = 6 \]
Thus, the value of \( b \) is:
\[ \boxed{6} \]