1. The problem is to understand the properties of the natural logarithm function $\ln$ and the exponential function $e^x$. 2. The key formulas are: - $\ln(e^x) = x$ - $e^{\ln x} = x$ 3. Explanation: - The natural logarithm $\ln$ is the inverse function of the exponential function $e^x$. - This means applying $\ln$ to $e^x$ returns the original exponent $x$. - Similarly, applying $e^x$ to $\ln x$ returns the original number $x$. 4. Important rules: - The domain of $\ln x$ is $x > 0$ because logarithms of non-positive numbers are undefined in real numbers. - The exponential function $e^x$ is defined for all real $x$. 5. These properties show that $\ln$ and $e^x$ "undo" each other, which is fundamental in solving equations involving exponentials and logarithms. Final answer: The expressions $\ln(e^x) = x$ and $e^{\ln x} = x$ demonstrate that $\ln$ and $e^x$ are inverse functions, meaning each reverses the effect of the other.