The expression you've uploaded is:

$\log_{\sqrt{6}} 216$

To solve this, we can use the change of base formula and simplify. Let's proceed step-by-step.

1. Rewrite the base: We know that $\sqrt{6} = 6^{1/2}$, so we can rewrite the expression as: $\log_{6^{1/2}} 216$

2. Use the Power Rule for Logarithms: The logarithmic identity $\log_{a^b} x = \frac{\log_a x}{b}$ applies here. Thus, we can rewrite this as: $\frac{\log_6 216}{1/2} = 2 \cdot \log_6 216$

3. Evaluate $\log_6 216$: We need to determine the value of $\log_6 216$. If we assume $216 = 6^x$, we find $x$ by solving the equation. However, $216$ is not an integer power of $6$, so we would likely need a calculator to approximate this logarithm in terms of base $10$ or base $e$.

Would you like me to continue with an approximation, or are you interested in additional details?

1. How do you convert between logarithmic bases in general?
2. What is the power rule for logarithms and when is it useful?
3. Why might approximations be necessary for non-integer powers in logarithmic calculations?
4. How can logarithmic identities simplify complex expressions?
5. How does understanding fractional exponents help in evaluating logarithmic expressions?

Tip: When dealing with unusual bases, rewriting in terms of simpler expressions can make calculations easier.