Standard Functions III — Exponential and Logarithmic Functions, with Graphs

Exponential function. For a fixed real number $a > 0$, $a \ne 1$, the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = a^x$ is called an exponential function with base $a$. Domain $= \mathbb{R}$ (a positive base raised to any real power is defined); range $= (0, \infty)$, since a positive base raised to any real power is always strictly positive, and every positive number is attained for some $x$. Two cases: when $a > 1$, $f(x) = a^x$ is strictly increasing (steadily climbing as $x$ increases, e.g. $f(x) = 2^x$ or the especially important natural exponential $f(x) = e^x$, using Euler's number $e \approx 2.71828$); when $0 < a < 1$, $f(x)=a^x$ is strictly decreasing instead (e.g. $f(x) = (1/2)^x$). In every case the graph passes through the fixed point $(0, 1)$, since $a^0 = 1$ for any valid base $a$, and never touches or crosses the $x$-axis, however far $x$ decreases (for $a>1$) or increases (for $0<a<1$) — the $x$-axis is a horizontal asymptote of the graph.

Logarithmic function. For the same base $a > 0$, $a \ne 1$, the logarithmic function $f(x) = \log_a x$ is defined as the inverse operation of the exponential: $y = \log_a x$ means precisely $a^y = x$. Because the exponential $a^y$ is only ever positive, $\log_a x$ is only defined for $x > 0$; domain $= (0, \infty)$; range $= \mathbb{R}$ (every real number is attained as $y$ ranges over all values, since $a^y$ sweeps through every positive number). As with the exponential, $\log_a x$ is strictly increasing when $a>1$ and strictly decreasing when $0 < a < 1$. Every logarithmic graph passes through the fixed point $(1, 0)$, since $\log_a 1 = 0$ for any valid base (as $a^0=1$), and the graph never crosses the $y$-axis, staying just to its right as $x \to 0^+$, where $\log_a x \to -\infty$ (for $a>1$) — the $y$-axis is a vertical asymptote. …