Use a change-of-base formula to evaluate the following logarithm. \[ \log _{7} \sqrt{13} \] Use the change-of-base formula to rewrite the given expression in terms of natural logarithms or common logarithms. \[ \log _{7} \sqrt{13}=\frac{\log \sqrt{13}}{\log 7} \] (Do not evaluate. Do hot simplify.) Evaluate the expression. \[ \log _{7} \sqrt{13}= \] (Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as needed.)

Step 1 ：Given the logarithm \(\log _{7} \sqrt{13}\), we can use the change-of-base formula to rewrite it in terms of natural logarithms. The change-of-base formula is \(\log_b a = \frac{\log a}{\log b}\).

Step 2 ：Applying the change-of-base formula, we get \(\log _{7} \sqrt{13} = \frac{\log \sqrt{13}}{\log 7}\).

Step 3 ：Next, we calculate the numerator and denominator separately. The numerator is \(\log \sqrt{13}\) and the denominator is \(\log 7\).

Step 4 ：Using a calculator, we find that the numerator is approximately 1.2824746787307684 and the denominator is approximately 1.9459101490553132.

Step 5 ：Dividing the numerator by the denominator, we get approximately 0.6590616115309205.

Step 6 ：Rounding to four decimal places as needed, the final answer is \(\boxed{0.6591}\).