Determine the correct evaluation of the expression e^(log|x|) and its behavior in mathematical contexts | Step-by-Step Solution

What We're Solving

We need to determine whether e^(log|x|) equals |x| or just x, and understand how this expression behaves mathematically, especially in contexts like differential equations.

The Approach

This is a question about the relationship between exponential and logarithmic functions! We'll use the fundamental property that exponential and logarithmic functions are inverses, but we need to be careful about domains and the absolute value. The key is understanding what happens when we have log|x| instead of just log(x).

Step-by-Step Solution

Step 1: Recall the basic inverse property For positive numbers, we know that e^(log(a)) = a when a > 0. This is because e^x and log(x) are inverse functions.

Step 2: Analyze the domain of log|x|

• log|x| is defined for all x ≠ 0 (both positive and negative x values)
• When x > 0: |x| = x, so log|x| = log(x)
• When x < 0: |x| = -x, so log|x| = log(-x)

Step 3: Evaluate e^(log|x|) for positive x When x > 0:

• log|x| = log(x)
• e^(log|x|) = e^(log(x)) = x
• Since x > 0, we also have |x| = x
• Therefore: e^(log|x|) = x = |x|

Step 4: Evaluate e^(log|x|) for negative x When x < 0:

• |x| = -x (which is positive!)
• log|x| = log(-x)
• e^(log|x|) = e^(log(-x)) = -x
• But -x is positive when x is negative, so -x = |x|
• Therefore: e^(log|x|) = |x|

Step 5: Consider the implications The expression e^(log|x|) always produces a positive result (or zero), regardless of whether x was originally positive or negative. This is because:

• The absolute value |x| is always non-negative
• The logarithm of a positive number exists
• The exponential of that logarithm gives us back the positive value

The Answer

e^(log|x|) = |x| for all x ≠ 0.

This is NOT equal to x when x is negative. The absolute value in the logarithm ensures that the result is always the absolute value of the original input.

Implications for differential equations:

• This property is crucial when solving differential equations where variables might take negative values
• It helps maintain continuity in solutions across zero
• It's particularly important in separation of variables when you get expressions like ∫(1/x)dx = log|x| + C

Memory Tip

Think of it this way: "The absolute value bars act like a filter - they ensure that only the 'size' of the number survives the exponential-logarithm roundtrip, not its sign!" The logarithm can only "remember" the magnitude, so when we exponentiate back, we only get the magnitude (absolute value) back.