So what exactly happens to the real interest rate in the long run? I have two books both written by MIT professors, Macroeconomics by Blanchard and Economics by Acemoglu.

Regarding to what happens to real interest rate in the medium/long run after the central bank lowers the interest rate, the two books has two very different explanations.

Blanchard explains through the IS-LM and AS-AD model that increase in real money stock leads to decrease in real and nominal interest rate in the short run, but in the medium run, because output returns to natural level of output, the real interest rate returns to the natural interest rate. Therefore money growth has no effect on real interest rate in the medium run.

But in Acemoglu's book it says and I quote


  The long-term real interest rate is the long-term nominal interest
  rate minus the long-term expected inflation rate. When the Fed
  influences short-term interest rates, such as the federal funds rate,
  this affects the long-term nominal interest rate. A long-term loan is
  like a combination of short-term loans. You can think of a 10-year
  loan as ten 1-year loans lined up one after the other. When the
  federal funds rate goes down, the interest rate for the first 1-year
  loan goes down. In addition, a change in the federal funds rate is
  usually not reversed for several years, so several of the 1-year loans
  in the 10-year loan package are affected. In most cases, when the Fed
  lowers the federal funds rate this has little impact on long-term
  inflationary expectations. Summing this up, the long-term real
  interest rate tends to fall when the federal funds rate falls because
  the long-term nominal interest rate falls and inflation expectations
  tend to stay roughly the same.


Did I misinterpret the two authors or what?

Can I assume Blanchard's medium run and Acemoglu's long-term are the same thing in this context?
Blanchard examines a case with a single interest rate, and what happens to it "in the medium long-run" (in the sense of the economy being in equilibrium). His discussion is the most abstract. 

Acemoglou discusses a more realistic case where many different interest rates exist at the same time, and analyzes the observed phenomenon that interest rates on short-term loans are different than interest rates on long-term loans, even when both contracts are entered into at the same calendar time: the time horizon of the contract matters for the interest rate itself. He then goes on to discuss how monetary policy may or may not affect expectations. 

So when Acemoglou says "the long-term real interest rate tends to fall..." he does not mean "the real interest rate characterizing long-run equilibrium tends to fall". He means "the real interest rate that is currently imposed on loans with a long-term horizon, tends to fall". 

These aspects are simply absent in the basic IS-LM / AS-AD analysis.