The image shows the expression:

$\log_{\sqrt{5}} 72.34$

To evaluate this, we can rewrite it using the change of base formula:

$\log_{\sqrt{5}} 72.34 = \frac{\log 72.34}{\log \sqrt{5}}$

Since $\log \sqrt{5} = \frac{1}{2} \log 5$, we substitute this into the equation:

$\log_{\sqrt{5}} 72.34 = \frac{\log 72.34}{\frac{1}{2} \log 5} = 2 \times \frac{\log 72.34}{\log 5}$

Now, we just need to compute the values of $\log 72.34$ and $\log 5$.

Let me calculate that.The value of $\log_{\sqrt{5}} 72.34$ is approximately 5.32.

Do you need further details or have any questions?

Here are 5 related questions you might explore:

1. What is the general formula for changing the base of a logarithm?
2. How does the square root in the base affect the logarithmic expression?
3. Can this technique be applied to any base involving roots or powers?
4. How would the result change if the base was $\log_5$ instead of $\log_{\sqrt{5}}$?
5. What are some practical applications of logarithms with non-integer bases?

Tip: Always simplify logarithmic expressions by applying known logarithmic identities before evaluating them numerically.