The value of a currency is how much people are willing to pay to have the other currency. For example, if Japan is importing many goods and services, they need to pay them in USD and so they have to buy USD on the market.
> Both, Japan and the US are increasing their monetary base by what, 10% per year?
The monetary base is not strictly connected to the money in the economy. Most of the money is created by commercial banks when they hand out loans. So the entire money in the economy can shrink (people paying back more loans than they), while at the same time the central bank increases currency.
For reference: https://www.bundesbank.de/resource/blob/654284/df66c4444d065...