40. Compute $\log _{4} 5$ using the change-of-base formula.

Step 1 ：Given the problem to compute \(\log _{4} 5\), we can use the change-of-base formula.

Step 2 ：The change-of-base formula is \(\log_b a = \frac{\log_c a}{\log_c b}\).

Step 3 ：We can use this formula to change the base from 4 to 10, as base 10 is the most common base and is easy to compute.

Step 4 ：So, we can rewrite \(\log_4 5\) as \(\frac{\log_{10} 5}{\log_{10} 4}\).

Step 5 ：Calculating \(\log_{10} 5\) gives approximately 0.6989700043360189.

Step 6 ：Calculating \(\log_{10} 4\) gives approximately 0.6020599913279624.

Step 7 ：Dividing these two values gives the value of \(\log_4 5\), which is approximately 1.1609640474436813.

Step 8 ：Final Answer: \(\boxed{1.1609640474436813}\)