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How does the natural exponential function e^x relate to the natural logarithm ln y?

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A graph of the function f(x) = x cubed minus 3x plus 2 is shown on a blue background, with a red curve passing through key points. Local maximum and minimum points are marked with black dots at x equals negative 1 and x equals 1, illustrating important features of the curve. — function Plot of the cubic equation f(x) = x³ − 3x + 2. The plotted points are where changes in curvature occur.

In mathematics, the natural exponential function $y = e^{x}$ and the natural logarithm function $\ln y$ are inverse functions.12

This relationship means that they effectively "undo" each other's actions. Specifically:

If $y = e^{x}$ , then $x = \ln y$ .31
Conversely, the composition of these functions returns the original input, such that $\ln (e^{x}) = x$ .3

The constant $e$ , approximately 2.71828, serves as the base for both the natural exponential function and the natural logarithm. Because they are inverses, their graphs are reflections of each other across the line $y = x$ . While the exponential function $e^{x}$ grows rapidly for all real numbers, the natural logarithm $\ln x$ is defined only for positive values of $x$ and represents the power to which the base $e$ must be raised to produce a given value.312