Simplify the logarithmic expression: Loge e2 + Log3 243 - Log625 25.

The natural logarithm $\text{Log}_e$ is denoted as $\ln$.
Using the property $\ln(e^x) = x$:

$$\text{Log}_e e^2 = \ln e^2 = 2$$

We need to express 243 as a power of 3.

$$3^5 = 243$$

So, using the property $\text{Log}_b b^x = x$:

$$\text{Log}_3 243 = \text{Log}_3 3^5 = 5$$

We need to express both 625 and 25 as powers of the same base.

$$25 = 5^2$$ $$625 = 25^2 = (5^2)^2 = 5^4$$

So the term becomes $\text{Log}_{5^4} 5^2$.
Using the property $\text{Log}_{b^k} A^m = \frac{m}{k} \text{Log}_b A$:

$$\text{Log}_{5^4} 5^2 = \frac{2}{4} \text{Log}_5 5$$

Since $\text{Log}_5 5 = 1$:

$$\frac{2}{4} \cdot 1 = \frac{1}{2}$$

$$2 + 5 - \frac{1}{2}$$ $$7 - \frac{1}{2}$$

To subtract, find a common denominator:

$$\frac{14}{2} - \frac{1}{2} = \frac{13}{2}$$

The simplified value of the expression is:

$$\frac{13}{2}$$