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Find the exact value of each expression.

(a) $ e^{-ln 2} $

(b) $ e^{\ln}^{(\ln e^3)} $

a) $\frac{1}{2}$

b) 3

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okay for this problem. We want to think about the inverse relationship of E to a power and the natural log functions. The each of the X function and the natural log of X function are in verses of each other. So when you do function composition one inside the other, they're going to cancel one another. So if you saw E to the natural log of X, it would just be X. And if you saw the natural log of each of the X, it would just be X. So let's take advantage of that to simplify these expressions. Looking at the one on the left, the first thing I want to do is use the power property of logarithms and bring this negative up and make a power of negative one. So now we hav e to the natural log of two to the negative first power. So we got the situation here. We have e to the natural log of something the e and the natural log function are going to cancel, leaving us with just two to the negative one, and to to the negative one is 1/2 for the other expression. We have a similar situation going on. We have e to the natural log of something and those functions air going to cancel, leaving you with just natural Aga v cubed. Well, here we have it again. We have the natural log of e to the something those functions air going to cancel, leaving you with just three.