The problem is stated as follows:

Given:
$\log a - \log b = -6$ Determine the value of:
$\log \sqrt[3]{\frac{b}{a}}$

Step-by-Step Solution

1. Simplify the given expression: We know the logarithmic property: $\log a - \log b = \log \left(\frac{a}{b}\right)$ So, the equation becomes: $\log \left(\frac{a}{b}\right) = -6$ This can be rewritten as: $\frac{a}{b} = 10^{-6}$ Therefore, $\frac{b}{a} = 10^6$.

2. Simplify the expression to be determined: Using the logarithmic property $\log \left( \sqrt[3]{x} \right) = \frac{1}{3} \log x$, the target expression becomes: $\log \sqrt[3]{\frac{b}{a}} = \frac{1}{3} \log \left( \frac{b}{a} \right)$

3. Substitute the value of $\frac{b}{a}$: We know $\frac{b}{a} = 10^6$, so: $\log \sqrt[3]{\frac{b}{a}} = \frac{1}{3} \log (10^6)$ Using $\log (10^6) = 6$, we get: $\frac{1}{3} \times 6 = 2$

Thus, the value of $\log \sqrt[3]{\frac{b}{a}}$ is 2.

Would you like further details or have any questions? Here are some related questions to explore:

1. How do logarithmic properties simplify equations?
2. Can you explain the steps to solving logarithmic equations?
3. What are other common logarithmic identities?
4. How does the base of the logarithm affect the solution?
5. How do we handle fractional or cube roots in logarithms?

Tip: When dealing with logarithms, always look out for opportunities to apply properties like the quotient rule, product rule, or power rule to simplify expressions.