Let's evaluate the given triple integral:

$I = \int_0^{\log 2} \int_0^x \int_0^{x+\log 2} e^{x+y+z} \, dz \, dy \, dx.$

Step 1: Compute the Inner Integral

The innermost integral is:

$\int_0^{x+\log 2} e^{x+y+z} \, dz.$

Since $e^{x+y+z}$ is independent of $z$, we integrate:

$\int_0^{x+\log 2} e^{x+y+z} \, dz = e^{x+y} \int_0^{x+\log 2} e^z \, dz.$

Evaluating the integral,

$\int e^z \, dz = e^z.$

So,

$e^{x+y} \left[ e^z \right]_0^{x+\log 2} = e^{x+y} \left( e^{x+\log 2} - e^0 \right).$

Since $e^{x+\log 2} = e^x e^{\log 2} = 2 e^x$, we get:

$e^{x+y} \left( 2e^x - 1 \right).$

Step 2: Compute the Middle Integral

Now, integrate with respect to $y$:

$\int_0^x e^{x+y} (2e^x - 1) \, dy.$

Since $(2e^x - 1)$ is constant with respect to $y$, we get:

$(2e^x - 1) e^x \int_0^x e^y \, dy.$

Evaluating,

$\int e^y \, dy = e^y.$

So,

$(2e^x - 1) e^x \left[ e^y \right]_0^x = (2e^x - 1) e^x \left( e^x - 1 \right).$

Expanding,

$(2e^x - 1) (e^{2x} - e^x) = 2e^{3x} - e^{2x} - 2e^{2x} + e^x.$

$= 2e^{3x} - 3e^{2x} + e^x.$

Step 3: Compute the Outer Integral

$\int_0^{\log 2} (2e^{3x} - 3e^{2x} + e^x) \, dx.$

Computing term by term:

1. $\int e^{3x} \, dx = \frac{e^{3x}}{3}$,
2. $\int e^{2x} \, dx = \frac{e^{2x}}{2}$,
3. $\int e^x \, dx = e^x$.

Evaluating from $0$ to $\log 2$:

1. $\left[ \frac{e^{3x}}{3} \right]_0^{\log 2} = \frac{e^{3\log 2}}{3} - \frac{e^0}{3} = \frac{2^3}{3} - \frac{1}{3} = \frac{8}{3} - \frac{1}{3} = \frac{7}{3}$,
2. $\left[ \frac{e^{2x}}{2} \right]_0^{\log 2} = \frac{e^{2\log 2}}{2} - \frac{e^0}{2} = \frac{2^2}{2} - \frac{1}{2} = \frac{4}{2} - \frac{1}{2} = \frac{3}{2}$,
3. $\left[ e^x \right]_0^{\log 2} = e^{\log 2} - e^0 = 2 - 1 = 1$.

Now compute:

$2 \times \frac{7}{3} - 3 \times \frac{3}{2} + 1.$

$\frac{14}{3} - \frac{9}{2} + 1.$

Convert to a common denominator (6):

$\frac{28}{6} - \frac{27}{6} + \frac{6}{6} = \frac{28 - 27 + 6}{6} = \frac{7}{6}.$

Final Answer:

$\frac{7}{6}.$

Would you like further clarifications? Here are some related questions:

1. How do you change the order of integration in triple integrals?
2. What are some real-world applications of triple integrals?
3. How do you evaluate triple integrals in spherical or cylindrical coordinates?
4. How does the Jacobian factor affect changing variables in multiple integrals?
5. What happens when the limits of integration are functions instead of constants?

Tip: Always check whether changing the order of integration simplifies the evaluation of a multiple integral.