When you receive your credit card statement (or bank statement if you used a debit card), you should see a line for the purchase expressed in the format
11,123 JPY @ 0.0094443 : $105.05. It really doesn't matter what rate your bank uses, they're going to collect from you in USD, thus your expense was incurred in USD.
In other words, you did not pay anybody in JPY. Your bank paid the merchant and then subtracted from your balance (or added to your debt) however much USD they felt was fair in exchange for the amount of JPY they just gave to the merchant.
Alternatively, if you had previously converted a sum of money into JPY banknotes or you have a JPY bank account that you fund periodically and later on paid for something using those funds/cash, that's when the exchange rate becomes more important.
It's important because the value of the JPY you exchanged for goods/services may have changed since the time you purchased it*, and we want to know its value at the time you spent the money, not the time you bought it. For that situation, the IRS says:
Translating foreign currency into U.S. dollars
You must express the
amounts you report on your U.S. tax return in U.S. dollars. Therefore,
you must translate foreign currency into U.S. dollars if you receive
income or pay expenses in a foreign currency. In general, use the
exchange rate prevailing (i.e., the spot rate) when you receive, pay
or accrue the item.
A spot rate is used when you aren't converting cash from one currency to another (because you already have the right one to settle), but you want to know what it would have cost you if you did need to buy the foreign currency at that moment in time to settle the transaction. The spot rate is determined by the demand for each currency based on the current bid and ask amounts on foreign exchange markets. There are plenty of websites you can reference to find out exactly how much of one currency you would have needed to buy an exact amount of another currency at any particular point in time.
Since there are multiple websites, whose spot rate should you use? Well, the IRS says:
Currency exchange rates
The Internal Revenue Service has no official
exchange rate. Generally, it accepts any posted exchange rate that is
When valuing currency of a foreign country that uses multiple exchange
rates, use the rate that applies to your specific facts and
So as long as you are consistent and don't flounder around a bunch of different sources looking for the "best" rate for each individual transaction, they don't mind, and neither should you.
Since they tell you to use the rate that applies to your specific facts and circumstances, it would make sense to use the spot rate provided by whomever sold you the JPY you are using for the transaction (if published/available).
*If you really want to do things by the book, you should also be reporting any change in the value of foreign currencies purchases as a capital gain/loss, but that's a whole different kettle of fish and beyond the scope of this question.