Exponential and Logarithmic Functions: eˣ and ln x

Core Concept

The Number $e$

The constant $e$ is an irrational number defined by:

It is the unique base for which the exponential function is its own derivative, i.e. $\dfrac{d}{dx}(e^x) = e^x$.

The Natural Exponential Function $y = e^x$

Key properties of $f(x) = e^x$:

• Domain: $x \in \mathbb{R}$ (all real numbers)
• Range: $y > 0$ (strictly positive for all inputs)
• $y$-intercept: $(0, 1)$, since $e^0 = 1$
• Asymptote: the $x$-axis ($y = 0$) is a horizontal asymptote as $x \to -\infty$
• Behaviour: strictly increasing; as $x \to +\infty$, $e^x \to +\infty$
• Always positive: $e^x > 0$ for all $x \in \mathbb{R}$

The Natural Logarithm $y = \ln x$

The natural logarithm is defined as the logarithm to base $e$:

Key properties of $g(x) = \ln x$:

• Domain: $x > 0$
• Range: $y \in \mathbb{R}$ (all real numbers)
• $x$-intercept: $(1, 0)$, since $\ln 1 = 0$
• Asymptote: the $y$-axis ($x = 0$) is a vertical asymptote as $x \to 0^+$
• Behaviour: strictly increasing; as $x \to +\infty$, $\ln x \to +\infty$ (slowly)

Inverse Function Relationship

$e^x$ and $\ln x$ are inverse functions of each other. This means:

Graphically, the curve $y = \ln x$ is the reflection of $y = e^x$ in the line $y = x$. Their domains and ranges are swapped: what is the domain of one is the range of the other.

The Graph of $y = e^{kx}$

For a constant $k \neq 0$, the function $y = e^{kx}$ is a horizontal stretch of $y = e^x$ by scale factor $\dfrac{1}{k}$.

In all cases: $y$-intercept is $(0, 1)$, the curve lies entirely above the $x$-axis, and $y = 0$ remains a horizontal asymptote.