Key Concepts and Formulas

• Functional Equation: An equation where the unknown is a function. Solving often involves substitution and differentiation.
• Differential Equations: Equations involving derivatives. Separable equations are of the form $\frac{dy}{dx} = g(x)h(y)$, solved by separating variables and integrating.
• Logarithm Properties: $\log_e(a^b) = b \log_e(a)$.
• Arithmetic Series: $\sum_{k=1}^n k = \frac{n(n+1)}{2}$.

Step-by-Step Solution

Step 1: Analyze the Given Information

We are given the functional equation $f(x+y) = f(x)f'(y) + f'(x)f(y) \quad (*)$ and the initial condition $f(0) = 1$. We need to find $\sum_{n=1}^{100} \log_e f(n)$.

Step 2: Find $f'(0)$

To find $f'(0)$, substitute $x=0$ and $y=0$ into the functional equation $(*)$: $f(0+0) = f(0)f'(0) + f'(0)f(0)$ $f(0) = 2f(0)f'(0)$ Since $f(0) = 1$, we have $1 = 2(1)f'(0)$ $f'(0) = \frac{1}{2}$

Step 3: Derive a Differential Equation

Substitute $y=0$ into the functional equation $(*)$: $f(x+0) = f(x)f'(0) + f'(x)f(0)$ $f(x) = f(x)f'(0) + f'(x)f(0)$ Substitute $f(0) = 1$ and $f'(0) = \frac{1}{2}$: $f(x) = f(x)\left(\frac{1}{2}\right) + f'(x)(1)$ $f(x) = \frac{1}{2}f(x) + f'(x)$ $f'(x) = \frac{1}{2}f(x)$

Step 4: Solve the Differential Equation

We have the differential equation $\frac{df}{dx} = \frac{1}{2}f(x)$ Separate the variables: $\frac{df}{f(x)} = \frac{1}{2}dx$ Integrate both sides: $\int \frac{df}{f(x)} = \int \frac{1}{2}dx$ $\ln|f(x)| = \frac{1}{2}x + C$ Exponentiate both sides: $|f(x)| = e^{\frac{1}{2}x + C} = e^{\frac{1}{2}x}e^C$ $f(x) = Ae^{\frac{1}{2}x}$ where $A = \pm e^C$ is a constant.

Step 5: Determine the Constant A

Use the initial condition $f(0) = 1$: $f(0) = Ae^{\frac{1}{2}(0)} = Ae^0 = A$ Since $f(0) = 1$, we have $A = 1$. Therefore, $f(x) = e^{\frac{x}{2}}$

Step 6: Evaluate the Summation

We need to find $S = \sum_{n=1}^{100} \log_e f(n) = \sum_{n=1}^{100} \log_e \left(e^{\frac{n}{2}}\right)$ Using the property $\log_e(e^k) = k$, we have $S = \sum_{n=1}^{100} \frac{n}{2} = \frac{1}{2} \sum_{n=1}^{100} n$ Using the formula for the sum of the first $n$ natural numbers, $\sum_{k=1}^n k = \frac{n(n+1)}{2}$, we get $S = \frac{1}{2} \left(\frac{100(100+1)}{2}\right) = \frac{1}{2} \left(\frac{100(101)}{2}\right) = \frac{1}{2} \left(\frac{10100}{2}\right) = \frac{10100}{4} = 2525$

Common Mistakes & Tips

• Remember to use the initial condition to find the constant of integration after solving the differential equation.
• Don't forget the properties of logarithms when simplifying the summation.
• Double-check your calculations, especially when dealing with fractions and summations.

Summary

We used the given functional equation and initial condition to derive a differential equation for $f(x)$. Solving this differential equation and applying the initial condition, we found $f(x) = e^{x/2}$. Then, we evaluated the sum $\sum_{n=1}^{100} \log_e f(n)$, which simplifies to $\frac{1}{2} \sum_{n=1}^{100} n$. Using the arithmetic series formula, we found the sum to be 2525.

The final answer is \boxed{2525}. This corresponds to option (C).