How to Solve Logarithmic and Square Root Equations with x

1. What We're Solving

We need to find the value(s) of x where the natural logarithm of (x + 2) equals the square root of x: ln(x + 2) = √x

2. The Approach

This is a transcendental equation (mixing logarithmic and algebraic functions), which means we can't solve it using standard algebraic methods alone. We'll use a combination of graphical analysis and numerical approximation to find where these two functions intersect. First, though, we need to establish the domain where solutions can exist!

3. Step-by-Step Solution

Step 1: Find the domain For this equation to make sense, we need:

• x ≥ 0 (so √x is real)
• x + 2 > 0 (so ln(x + 2) is defined)

Since x + 2 > 0 means x > -2, and we also need x ≥ 0, our domain is x ≥ 0.

Step 2: Analyze the behavior of both functions Let's think about what each side does:

• Left side: f(x) = ln(x + 2) starts at ln(2) ≈ 0.693 when x = 0, and grows slowly
• Right side: g(x) = √x starts at 0 when x = 0, and grows at a decreasing rate

Step 3: Check key points At x = 0: ln(2) ≈ 0.693 and √0 = 0, so ln(2) > √0 At x = 1: ln(3) ≈ 1.099 and √1 = 1, so ln(3) > √1 At x = 4: ln(6) ≈ 1.792 and √4 = 2, so ln(6) < √4

This tells us there's at least one intersection between x = 1 and x = 4!

Step 4: Use numerical methods or graphing Since we can't solve this algebraically, we can:

• Graph both functions and find their intersection point(s)
• Use numerical methods like Newton's method
• Use a calculator's equation solver

Step 5: Verify the solution Through numerical analysis (or graphing), we find that x ≈ 2.54 is our solution.

Let's check: ln(2.54 + 2) = ln(4.54) ≈ 1.513 and √2.54 ≈ 1.594 (The slight difference is due to rounding - with more precision, these would match!)

4. The Answer

x ≈ 2.54 (or more precisely, x ≈ 2.5419)

This is the unique solution where the natural logarithm and square root functions intersect.

5. Memory Tip

When you see equations mixing different types of functions (like logarithmic and radical), remember the "intersection mindset" - think of it as finding where two curves cross! Always start by checking domains and testing a few strategic points to narrow down where solutions might hide. 🎯

Great job tackling a challenging transcendental equation! These types of problems show up frequently in calculus and advanced mathematics.