Workout: Why entropy is logarithmic

Read

Read the page Why entropy is logarithmic and answer the following questions.

Answer these questions

1. That page makes the claim that you can turn multiplication into addition by taking the logarithm. Confirm that this is correct by taking two pairs of two-digit numbers and multiplying them. For each case show that the sum of the logarithms of the two terms that are multiplied is equal to the logarithm of the product. Does it depend on whether you are using log to the base e (natural log = ln) or to the base 10?

2. To see why this is relevant to our analysis of the arrangement of energy in degrees of freedom, first take a simple problem in the ways that you can arrange some cards from a deck. 

a. Suppose you have only the numbered cards from one suit (say hearts) of one deck (numbered from Ace = 1 to 10). If you choose 4 of those cards at random, how many different sets of those cards could you get? (Order doesn't matter, so 10-9-8-7 is considered the same as 7-8-9-10.)

b. If you now select from the numbered cards from a different suit (say spades) of that deck and choose 3 of those cards at random, how many different sets of the spades could you get?

c. If you select BOTH four hearts and three spades at random, how many different sets of cards could you get?

3. Our job in studying how energy is distributed will often involve systems that have multiple parts, for example, two boxes of different temperatures in contact. Why do you think taking logarithms might have value in considering how energy could be arranged in the two systems?

Joe Redish 1/28/18