Observe that the inside integral cannot be expressed in terms of elementary functions.change the order of integration and so evaluate the double integral. $$ \int_{x=0}^{\ln 16} \int_{y=e^{2}}^{4} \frac{d y d x}{\ln y} $$

In double integrals, the order of integration refers to the sequence in which you integrate with respect to each variable. In our exercise, the original order is \(\begin{equation*} \int_{x=0}^{\text{ln} 16} \int_{y=e^{2}}^{4} \frac{d y d x}{\text{ln} y} \end{equation*}\). This means we first integrate with respect to \(\text{y}\) and then with respect to \(\text{x}\).
Changing the order of integration can sometimes simplify evaluation, especially when the inner integral cannot be expressed in terms of elementary functions.
To change the order, you need to reverse the integration limits and rewrite the integral accordingly. This often involves reinterpreting the region of integration in terms of different variables.

The region of integration is defined by the bounds on the integral. In our exercise, this region is given by the inequalities \(0 \leq x \leq \text{ln} 16\) and \(e^{2} \leq y \leq 4\).
When changing the order of integration, it's crucial to accurately express the same region using different inequalities. This involves translating bounds such as \(0 \leq x \leq \text{ln} 16\) to \(0 \leq x \leq \text{ln} y\) and adjusting accordingly.
In our case, we swap x and y to get \(e^2 \leq y \leq 4\) and \(0 \leq x \leq \text{ln} y\), transforming the integral for easier computation.

The inner integral is the first integral to be evaluated based on the adjusted limits. After changing the order of integration in our exercise, we get: \(\begin{equation*} \int_{y=e^{2}}^{4} \int_{x=0}^{\text{ln} y} \frac{d x d y}{\text{ln} y} \end{equation*}\).
The goal here is to simplify the expression inside the integral as much as possible before integrating. We note that we are integrating \(1 / \text{ln} y\) with respect to x, which simplifies to:
\(\begin{equation*} \int_{0}^{\text{ln} y} d x = [x]_{0}^{\text{ln} y} = \text{ln} y - 0 = \text{ln} y \end{equation*}\). This simplifies the problem tremendously.

Once the inner integral is evaluated, we move on to the outer integral. With our simplified inner integral, our double integral now looks like: \(\begin{equation*} \int_{y=e^{2}}^{4} \frac{\text{ln} y d y}{\text{ln} y } \end{equation*}\).
Notice here that \(\text{ln} y\) cancels out, leaving us with:
\(\begin{equation*} \int_{y=e^{2}}^{4} d y = [y]_{ e^{2}}^{ 4} = 4 - e^{2} \end{equation*}\).
The evaluation of the outer integral completes the solution, giving us the final answer: \(4 - e^{2}\).