the dollar can’t lose value against the Euro, gold, oil, etc., while at the same time we wouldn’t experience inflation at home. You should look up “purchasing power parity” as a basic concept.
In the short term, a $3 burger may still be $3 while the Euro, gold, etc. go up in value: But such divergence is unstainable, sooner or later that burger (and cars, clothes, etc.) will cost a lot more than $3.
Why does the burger cost $3? Certainly not because of commodity prices. At todays sky-high prices, one pound of high-quality wheat flour costs around 50 cents, and you can make 10-15 hamburger buns from that flour. So, one burger contains as much as 3-5c worth of flour (up from 1-2c in 2006!) 5c of cheese, 5c of beef, some lettuce and mustard, etc. The bulk is LABOR.
Even though dollar goes down, financial capacity of burger buyers stays the same. Dollar-denominated prices of flour and beef go up (because you can sell them overseas), therefore, the part that gets squeezed is labor. You have fewer cashiers, burger makers, and truck drivers working for less money making slightly fewer slightly more expensive burgers. To compensate, farmers, aircraft manufacturers, and everyone else who’s paid in Euros becomes slightly richer. Some people quit cashier jobs and become farm-workers. A new equilibrium is reestablished where tradable goods (flour) are more expensive than before, and nontradable goods (truck driver services) are cheaper than before.