We are tasked with solving for $b$ given:

$\log_b 6 = \frac{1}{3}$

Step-by-step Solution:

1. Understand the logarithmic form:

   The equation $\log_b 6 = \frac{1}{3}$ can be rewritten in its exponential form as:

   $b^{\frac{1}{3}} = 6$

   This means the cube root of $b$ equals 6.

2. Eliminate the cube root:

   To solve for $b$, cube both sides of the equation:

   $\left(b^{\frac{1}{3}}\right)^3 = 6^3$

3. Simplify:

   • The left-hand side simplifies to $b$, since the cube and the cube root cancel.
   • The right-hand side becomes $6^3 = 216$.

   Thus:

   $b = 216$

Final Answer:

$b = 216$

Let me know if you need further clarification or a detailed explanation of any step!

Related Questions:

1. How do you convert a logarithmic equation into exponential form?
2. If $\log_b x = y$, how do you solve for $b$ or $x$?
3. What are the properties of logarithms (e.g., product, quotient, and power rules)?
4. How do you solve for $x$ in an equation like $b^x = y$?
5. What is the meaning of a fractional logarithmic value like $\log_b a = \frac{1}{n}$?

Tip: To solve logarithmic equations, always consider rewriting the equation in its exponential form for easier manipulation.