You should:
Define an appropriate metric of performance (e.g., responsiveness, throughput, etc.). Then you should measure this metric with all logging turned off and then on. The difference would be the cost of logging.
Then you should experiment with different logging libraries and the modes they provide and document the observed differences.
In my personal experience, for all the three projects I worked on, I found that asynchronous logging helped improve the application throughput a lot. But the same may not hold for you, so make sure you make your decision after careful measurements.
The following does not directly relate to your question.
I noticed that you specifically mentioned business logging. In this case, you may also want to keep logging relevant and clean, in case you find your log files are growing huge and difficult to understand. There is a generally accepted design pattern in this area: log as per function. This would mean that business logging (e.g., customer requested a refund) goes to a different destination, interface logging would go to another destination (e.g., user clicked the upvote button != user upvoted an answer), and a cross system call would go to another destination (e.g., Requesting clearance through payment gateway). Some people keep a master log file with all events as well just to see a timeline of the process while some design log miners/scrappers to construct timelines when required.
Hope this helps,