1. The problem asks to explain why $$e^{\ln|x|} = |x|$$ holds true. 2. Recall the definition of the natural logarithm function $\ln y$ as the inverse of the exponential function $e^y$. This means: $$e^{\ln y} = y$$ for any positive number $y > 0$. 3. Since $|x|$ is always positive (or zero, but zero is excluded because $\ln 0$ is undefined), we can apply this property by substituting $y = |x|$: $$e^{\ln |x|} = |x|$$ 4. This equality holds because the exponential and natural logarithm functions are inverses, so applying one after the other returns the original input. 5. In summary, the expression $$e^{\ln |x|} = |x|$$ is true by the fundamental inverse relationship between $e^x$ and $\ln x$, and the absolute value ensures the argument of $\ln$ is positive, making the expression valid.