Exponential Functions with Different Bases

Definition Let $a$ be a real number with $a > 0$ and $a \ne 1$. The exponential function with base $a$ is

More generally, a function $f(x) = C a^x$ with a constant $C \ne 0$ is called an exponential function with base $a$ and initial value $C$. For integer $x$ the power $a^x$ is repeated multiplication or division, for rational $x = p/q$ with $q > 0$ it is $\left(\sqrt[q]{a}\right)^p$, and for irrational $x$ it is the value that the rational powers close in on. If $a > 1$ the function $a^x$ is strictly increasing, and if $0 < a < 1$ it is strictly decreasing. The base $a = 1$ is excluded because $1^x = 1$ is the constant function, a separate case. Negative bases are excluded because powers such as $(-2)^{1/2}$ are not real numbers.

Definition If $a > 1$, the base $a$ of $f(x) = C a^x$ is called the growth factor. With $C > 0$ the function is strictly increasing on $\mathbb{R}$, which means $x_1 < x_2$ implies $f(x_1) < f(x_2)$. Writing $a = 1 + r$ with $r > 0$, the number $r$ is the growth rate per unit of input, often given as a percent.

Definition If $0 < a < 1$, the base $a$ of $f(x) = C a^x$ is called the decay factor. With $C > 0$ the function is strictly decreasing on $\mathbb{R}$. Writing $a = 1 - r$ with $0 < r < 1$, the number $r$ is the fraction lost per unit of input. Since

the graph of $y = (1/a)^x$ is the reflection of the graph of $y = a^x$ across the $y$-axis.

Definition For $f(x) = C a^x$ and any fixed increment $\Delta$, every input $x$ satisfies

The ratio depends only on the increment $\Delta$, not on the starting input $x$. Equally spaced inputs therefore give outputs that form a geometric sequence. A linear function has the matching property for differences, since $(mx + m\Delta + b) - (mx + b) = m\Delta$.

Definition For $a > 0$ with $a \ne 1$,

So the range of $a^x$ is $(0, \infty)$ and its graph crosses the $y$-axis at $(0, 1)$. For $f(x) = C a^x$ the $y$-intercept is $(0, C)$, and every output has the same sign as $C$.

Definition A line $y = L$ is a horizontal asymptote of the graph of $f$ if $f(x)$ approaches $L$ as $x \to \infty$ or as $x \to -\infty$. For $a > 1$ the values $a^x$ approach $0$ as $x \to -\infty$, and for $0 < a < 1$ they approach $0$ as $x \to \infty$. In both cases the line $y = 0$ is a horizontal asymptote on one side only. For

the horizontal asymptote is $y = k$, and the graph never reaches it because $C a^x \ne 0$.

Definition The contextual domain of a model is the set of inputs for which the model is meant to describe the situation. It is a subset of the domain of the formula. For an exponential model the formula accepts every real number, but a stated time window such as $0 \le t \le T$ restricts the domain to $[0, T]$, and the range shrinks to the outputs on that interval. When the quantity is measured only at whole time units, the domain may even be a finite set such as $\{0, 1, 2, \dots, T\}$.