Evaluate the given logarithmic expression below by using the change of base formula (round your answer to the nearest thousandth). \[ \log _{6} 13 \] 1.187 1.287 1.432 1.247

Step 1 ：We are given the logarithmic expression \(\log _{6} 13\) and we are asked to evaluate it using the change of base formula.

Step 2 ：The change of base formula is a mathematical formula used to change the base of a logarithm. The formula is \(\log_b a = \frac{\log_c a}{\log_c b}\).

Step 3 ：We can use the change of base formula to change the base to 10 (since most calculators use base 10), so we have \(\log_6 13 = \frac{\log_{10} 13}{\log_{10} 6}\).

Step 4 ：Calculating \(\log_{10} 13\) and \(\log_{10} 6\), we get approximately 1.1139433523068367 and 0.7781512503836436 respectively.

Step 5 ：Substituting these values into the formula, we get \(\log_6 13 = \frac{1.1139433523068367}{0.7781512503836436} = 1.432\) when rounded to the nearest thousandth.

Step 6 ：Final Answer: The value of \(\log _{6} 13\) rounded to the nearest thousandth is \(\boxed{1.432}\).