Use the change-of-base theorem to find the logarithm.
\[
\log _{\sqrt{13}} 20
\]

Step 1 ：Given the logarithm \(\log _{\sqrt{13}} 20\)

Step 2 ：We can use the change-of-base theorem to express this in terms of natural logarithms (base e). The change-of-base theorem states that for any positive numbers a, b, and c, where a ≠ 1 and b ≠ 1, the logarithm base b of a can be computed with the formula: \[\log_b a = \frac{\log_c a}{\log_c b}\]

Step 3 ：So, we can express \(\log _{\sqrt{13}} 20\) as \(\frac{\ln 20}{\ln \sqrt{13}}\)

Step 4 ：Calculate the numerator \(\ln 20\), which is approximately 2.996

Step 5 ：Calculate the denominator \(\ln \sqrt{13}\), which is approximately 1.282

Step 6 ：Divide the numerator by the denominator to get the value of \(\log _{\sqrt{13}} 20\), which is approximately 2.336

Step 7 ：Final Answer: The value of \(\log _{\sqrt{13}} 20\) is approximately \(\boxed{2.336}\)