To approximate $\log_{\sqrt{2}}(6.5)$ using the change-of-base formula, we can use the following formula:

$\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$

where $b = \sqrt{2}$ and $a = 6.5$, and $c$ is typically 10 or $e$ for convenience (we will use $e$ for natural logarithms).

Using natural logarithms ($\ln$):

$\log_{\sqrt{2}}(6.5) = \frac{\ln(6.5)}{\ln(\sqrt{2})}$

First, let's find the natural logarithms of 6.5 and $\sqrt{2}$:

$\ln(6.5) \approx 1.8718$

$\ln(\sqrt{2}) = \ln(2^{1/2}) = \frac{1}{2} \ln(2) \approx \frac{1}{2} \times 0.6931 \approx 0.3466$

Now, we can calculate the logarithm:

$\log_{\sqrt{2}}(6.5) = \frac{1.8718}{0.3466} \approx 5.4001$

Thus, the value of $\log_{\sqrt{2}}(6.5)$ accurate to the nearest ten-thousandth is approximately $5.4001$.

Do you want details or have any questions?

Here are five numbered questions for you to ask:

1. How do you find natural logarithms of numbers?
2. Can you explain why we use the change-of-base formula?
3. What are other methods to calculate logarithms besides using the change-of-base formula?
4. How do logarithmic properties apply to solving equations?
5. Can you demonstrate how to use logarithms in a real-world problem?

Tip: When working with logarithms, remember that the change-of-base formula is a powerful tool for simplifying calculations, especially when the base is not one of the common logarithm bases (10 or $e$).