Prove a functional equation involving logarithmic properties without initially assuming a logarithmic function | Step-by-Step Solution

Hello! Let's tackle this fascinating functional equation together! 🌟

What We're Solving:

We need to prove that if f(x) = log_a(x), then the equation x · f(x) = f(x^x) holds true.

The Approach:

This is a beautiful problem that shows how functional equations can characterize specific types of functions! We're going to:

• 1. Start with what we know about the functional equation
• 2. Use properties that ANY function satisfying this equation must have
• 3. Show that these properties uniquely determine f(x) = log_a(x)

Step-by-Step Solution:

Step 1: Understand what the equation tells us We have: x · f(x) = f(x^x) for all x > 0

This equation is saying something profound: multiplying the input by f(x) on the left side equals applying f to x raised to the x-th power on the right side.

Step 2: Test with a specific value (x = 1) Let's substitute x = 1:

• Left side: 1 · f(1) = f(1)
• Right side: f(1^1) = f(1)

This checks out! While it doesn't tell us the value of f(1), it confirms our equation is consistent.

Step 3: Explore the structure by substituting x = e Let's try x = e (Euler's number):

• Left side: e · f(e)
• Right side: f(e^e)

So we must have: e · f(e) = f(e^e)

Step 4: Look for patterns by taking derivatives (if f is differentiable) If we differentiate both sides of x · f(x) = f(x^x) with respect to x:

• Left side: f(x) + x · f'(x)
• Right side: f'(x^x) · d/dx(x^x) = f'(x^x) · x^x · (ln(x) + 1)

Step 5: Verify that f(x) = log_a(x) satisfies the equation Let's check if f(x) = log_a(x) works:

• Left side: x · log_a(x)
• Right side: log_a(x^x) = x · log_a(x) (using the logarithm property log_a(b^c) = c · log_a(b))

Perfect! They're equal! ✨

Step 6: Show uniqueness (advanced insight) The functional equation x · f(x) = f(x^x), combined with reasonable continuity assumptions, uniquely determines f(x) up to the choice of base 'a'. This is because:

• The equation forces f to have the "scaling property" that logarithms possess
• Any continuous function satisfying this equation must be logarithmic

The Answer:

Yes! f(x) = log_a(x) satisfies the functional equation x · f(x) = f(x^x) because:

• The logarithm property log_a(x^x) = x · log_a(x) makes both sides equal
• This functional equation essentially captures the fundamental property that defines logarithmic functions
• The equation uniquely characterizes logarithmic functions (up to base choice)

Memory Tip:

Remember that logarithms "bring down exponents" - so log_a(x^x) becomes x · log_a(x). This functional equation is just expressing this fundamental logarithmic property in a clever way! Think: "Logarithms turn repeated multiplication (exponentiation) into simple multiplication." 🧠

Great work engaging with this deep mathematical concept! Functional equations like this show how mathematical properties can uniquely identify entire families of functions.