Logarithm of exponential

Since Log is a multi-valued inverse function of Exp, Mathematica doesn't evaluate Log[Exp[a]] (or equivalently, Log[E^a]). If you want to simplify these, you need to provide an assumption on the domain of a, e.g.,

Simplify[Log[E^a], a ∈ Reals]

a

or

Simplify[Log[E^a], a > 0]

a

as suggested in the other answer. Another method is to use PowerExpand:

PowerExpand[Log[E^a], Assumptions -> True]

a + 2 I π Floor[1/2 - Im[a]/(2 π)]

Addendum

As an aside, Log behave exactly like ArcSin here:

ArcSin[Sin[x]]

ArcSin[Sin[x]]

Including a domain restriction:

Simplify[ArcSin[Sin[x]], -Pi/2 < x < Pi/2]

x

For larger domains, Simplify doesn't work:

Simplify[ArcSin[Sin[x]], 0 < x < 2 Pi]

ArcSin[Sin[x]]

Again, using PowerExpand is useful:

p = PowerExpand[ArcSin[Sin[x]], Assumptions -> 0 < x < 2Pi];
p //TeXForm

$\begin{cases} \pi -x & \frac{\pi }{2}<x\leq \frac{3 \pi }{2} \\ x & x\leq \frac{\pi }{2} \\ x-2 \pi & \text{True} \end{cases}$

Visualization:

GraphicsColumn[{
    Plot[ArcSin[Sin[x]], {x, 0, 2 Pi}],
    Plot[p, {x, 0, 2 Pi}]
}]

The assumption a > 0 is needed when Simplify is called:

Assuming[a > 0, Log[Exp[a]] // Simplify]

a