Does paying by debit card on amazon.com show your purchase on your bank statement?

It shows where you shopped. It doesn't show what you bought. If you purchased the item through Amazon, the statement will only say "amazon.com".

It will not cost you any more than if you had paid in cash and will show on your bank statement just like every other debit transaction. Some stations have a pre-paid hold of some type. One time when I purchased gas and then checked my account online immediately after it showed a $75.00 hold withdrawal and then the $43.00 actual charge. I thought for sure someone at the gas station was trying to steal funds from customers accounts. I called and talked to the store manager and he informed me that it would simply disappear from my account once the real charge processed and he was right, and there was nothing on my bank statement.

yes, all transactions will be logged.