Determine whether the following statements are true and give an explanation or counterexample.
(4x+1)^{ln x} = x^ln(4x+1)

Verified step by step guidance

1. Step 1: Start by examining the given equation: \((4x+1)^{\ln x} = x^{\ln(4x+1)}\). We need to determine if this equation holds true for all values of \(x\) or if there is a counterexample.
2. Step 2: Consider taking the natural logarithm of both sides of the equation to simplify the exponents. This gives us \(\ln((4x+1)^{\ln x}) = \ln(x^{\ln(4x+1)})\).
3. Step 3: Apply the logarithmic identity \(\ln(a^b) = b \cdot \ln a\) to both sides. This results in \(\ln x \cdot \ln(4x+1) = \ln(4x+1) \cdot \ln x\).
4. Step 4: Notice that both sides of the equation are identical, \(\ln x \cdot \ln(4x+1) = \ln x \cdot \ln(4x+1)\), which suggests that the original equation is true for all \(x > 0\) where the logarithms are defined.
5. Step 5: Conclude that the statement is true for all \(x > 0\) where both sides of the equation are defined, as the simplification shows both sides are equal.