The Natural Logarithm Function

Definition The natural logarithm is the logarithm with base $e$,

So for $x > 0$, $\ln x$ is the unique real number $y$ with $e^y = x$. In symbols,

The base $e \approx 2.71828$ is positive and not equal to $1$, so everything true of $\log_a x$ for a base $a > 1$ is true of $\ln x$.

Definition The functions $e^x$ on $\mathbb{R}$ and $\ln x$ on $(0, \infty)$ are inverses of each other, which means

The graph of $y = \ln x$ is the reflection of the graph of $y = e^x$ across the line $y = x$. It is strictly increasing on $(0, \infty)$, and its range is $\mathbb{R}$.

Definition The real function $\ln x$ is defined exactly for $x > 0$. Zero and negative inputs have no real natural logarithm, because $e^y > 0$ for every real $y$. For $\ln(g(x))$ the domain is the set of $x$ in the domain of $g$ with $g(x) > 0$.

Definition Since $e^0 = 1$ and $e^1 = e$,

The point $(1, 0)$ is the only $x$-intercept of $y = \ln x$, and the point $(e, 1)$ marks where the graph reaches height $1$.

Definition For the natural logarithm,

A negative output is perfectly valid. What is not allowed is a nonpositive input.

Definition For a real constant $h$, the function

is defined exactly when $x > h$, so its domain is $(h, \infty)$. Its graph is the graph of $y = \ln x$ shifted $h$ units to the right when $h > 0$, and $\lvert h \rvert$ units to the left when $h < 0$. It has vertical asymptote $x = h$ and $x$-intercept $(h + 1, 0)$. Adding a constant $k$, as in $y = \ln(x - h) + k$, moves the graph up by $k$ and does not change the domain or the asymptote.

Definition As $x \to 0^+$, $\ln x \to -\infty$, so the line $x = 0$ is a vertical asymptote of $y = \ln x$. For $y = \ln(x - h)$, $\ln(x - h) \to -\infty$ as $x \to h^+$, so the line $x = h$ is the vertical asymptote. The graph approaches it only from the right and never meets it.