Exponential functions: the shape before the algebra

An exponential function has a fixed positive base raised to a variable power. In this topic, we write

$$f(x)=a^x$$

Here, $f(x)$ is the function’s output, $a$ is the positive base and $x$ is the real input variable. For the usual non-constant exponential graphs, take $a>0$ and $a\neq 1$. One special exponential function is

$$f(x)=e^x$$

Its base is $e$, the mathematical constant approximately equal to $2.718$.

The domain of $f(x)=a^x$ is $\mathbb{R}$, since any real value of $x$ can be used. Its range is $f(x)>0$: raising a positive base to a real power never produces zero or a negative value. Because $a^0=1$, the graph always passes through $(0,1)$. The line $y=0$ is a horizontal asymptote.

An asymptote is a straight line that a graph approaches increasingly closely without meeting in the part of the graph being described. For $a^x$, the horizontal asymptote $y=0$ shows that exponential decay can become very small, but the pure model never actually reaches zero.

The value of $a$ determines the graph’s direction. When $a>1$, the function increases: $a^x$ grows as $x$ increases. When $0<a<1$, it decreases, so $a^x$ decays as $x$ increases. In both cases, the intercept is still $(0,1)$ and the horizontal asymptote is still $y=0$.

In a model, $a>1$ represents repeated multiplication by a factor greater than one, so it can describe growth. Compound interest and geometric sequences fit this idea naturally because each time period multiplies the previous amount by a fixed factor. A base with $0<a<1$ instead represents repeated multiplication by a factor between zero and one, as in radioactive decay or discharging capacitors. Take care with the everyday phrase “exponential growth”. People often use it to mean “very fast growth”, but mathematically it describes growth by a constant multiplicative factor over equal input intervals.

Why $e^x$ is the standard exponential

Every positive-base exponential can be rewritten using $e$. The key identity is

$$a^x=e^{x\ln a}$$

where $\ln a$ is the natural logarithm of the base $a$. This explains why $e^x$ appears so often in calculus, statistics, finance, physics, chemistry and biology. Changing the base simply changes the multiplier of $x$ in the exponent.

A common form used in modelling is

$$y=Ae^{kx}$$

where $y$ is the modelled output, $A$ is the initial value when $x=0$, $k$ is the continuous growth or decay constant and $x$ is the input variable. When the context includes units, $x$ carries those units and $k$ has reciprocal units, making $kx$ dimensionless. For example, if $x$ is time in seconds, then $k$ is measured in $\text{s}^{-1}$. The model grows if $k>0$ and decays if $k<0$.

Consider the population model $P(t)=P_0e^{kt}$. Here, $P(t)$ is the population at time $t$, $P_0$ is the initial population, $k$ is the growth constant and $t$ is time. It’s a simplified description rather than reality itself. Biological growth may begin exponentially, but limited resources usually stop that pattern from continuing forever. A mathematical model captures selected features of a situation while deliberately leaving others out.

Logarithmic functions: undoing exponentials

A logarithmic function gives the exponent needed to produce a positive input from a fixed positive base. We write

$$f(x)=\log_a x$$

where $f(x)$ is the output exponent, $a$ is the base with $a>0$ and $a\neq 1$, and $x$ is the positive input. The restriction $x>0$ is essential. Logarithms of zero and negative numbers aren’t defined in the real-number graphing used here.

The base is the fixed positive number repeatedly multiplied in an exponential expression. In $\log_a x$, the base $a$ identifies the exponential function being undone. The natural logarithm has base $e$ and is written

$$f(x)=\ln x$$

where $x>0$.

The graph of $y=\ln x$ has domain $x>0$ and range $\mathbb{R}$. Since $\ln 1=0$, it passes through $(1,0)$, and the line $x=0$ is a vertical asymptote. The graph increases slowly. Logarithms keep increasing, but every additional equal rise in output requires a multiplicative increase in input.

For $a>1$, the graph of $y=\log_a x$ has the same general increasing shape as $y=\ln x$. When $0<a<1$, the logarithmic graph decreases. This matches the exponential case: bases greater than one produce increasing exponential and logarithmic functions, whereas bases between zero and one produce decreasing functions.

Change of base

The change-of-base formula is

$$\log_a x=\frac{\ln x}{\ln a}$$

where $a$ is the logarithm base with $a>0$ and $a\neq 1$, and $x$ is the positive input. This identity appears in the formula booklet, but it’s worth understanding rather than merely quoting. It shows that any logarithm can be expressed as a vertical scale change of $\ln x$.

If $a>1$, then $\ln a>0$, making $\log_a x$ a positive multiple of $\ln x$. If $0<a<1$, then $\ln a<0$, so the graph is also reflected in the $x$-axis. The symbolic formula therefore gives the graph transformation directly—an example of equivalent representations.

Exponentials and logarithms are inverse functions

Inverse functions reverse each other’s input-output mapping. Exponential and logarithmic functions are inverses because each undoes the other:

$$\log_a(a^x)=x$$

where $a$ is the positive base with $a\neq 1$, and $x$ is a real input, and

$$a^{\log_a x}=x$$

where $x>0$.

On a graph, inverse functions are reflections of one another in the line $y=x$. The graph of $y=\log_a x$ is therefore the reflection of $y=a^x$ in $y=x$. Their domains and ranges swap: $y=a^x$ has domain $\mathbb{R}$ and range $y>0$, while $y=\log_a x$ has domain $x>0$ and range $\mathbb{R}$.

This inverse relationship is the main reason logarithms can solve exponential equations. For example, if

$$a^x=b$$

where $a>0$, $a\neq 1$ and $b>0$, then applying $\log_a$ gives

$$x=\log_a b$$

or, using natural logarithms,

$$x=\frac{\ln b}{\ln a}$$

where $b$ is the positive output value being reached.

Reading and using models

Exponential and logarithmic functions appear across many subjects because they describe relationships involving multiplication, scaling and rates of proportional change. Examples include radioactive decay, capacitor discharge, first-order chemical reactions, activation energy relationships, compound interest and some biological growth curves. Each can lead naturally to an exponential or logarithmic representation.

One useful modelling technique is to take logarithms, turning a power or exponential relationship into a straight-line relationship. Suppose, for example, that

$$y=Cx^n$$

where $y$ is the output quantity, $C$ is a positive constant multiplier, $x$ is the positive input quantity and $n$ is the power. Taking natural logarithms gives

$$\ln y=\ln C+n\ln x$$

If $\ln y$ is plotted against $\ln x$, this has the form of a straight line. Its gradient is $n$ and its vertical intercept is $\ln C$. Logarithms aren’t just a calculator button; they change the representation so that the relationship becomes easier to see.

Whenever you use one of these models, check its assumptions. A compound-interest model assumes a fixed percentage rate, while a radioactive-decay model assumes a constant decay probability. A biological growth model may ignore environmental limits. Mathematics supplies a clean structure; the context determines how far that structure can be trusted.