$\log _{15} 89.5$

Step 1 ：The problem is asking for the logarithm base 15 of 89.5. This can be calculated directly using the formula for calculating a logarithm with a base other than e (the natural logarithm): \[\log_b a = \frac{\log a}{\log b}\] where \(\log\) is the natural logarithm. In this case, \(a = 89.5\) and \(b = 15\).

Step 2 ：Substitute the values into the formula: \[\log_{15} 89.5 = \frac{\log 89.5}{\log 15}\]

Step 3 ：Calculate the value: \[\log_{15} 89.5 = 1.6595846795792777\]

Step 4 ：Round the final answer to two decimal places: \[\boxed{1.66}\]