Relationship of the functions f(x) = log_ax and g(x) = a^x

If $a$ is any positive number, then the functions $f(x) = \log_ax$ and $g(x) = a^x$ are inverse functions.

This means that

$a^{\log_ax} = x$         and         $\log_a(a^x) = x$

for every real number $x.$

Want to learn more about inverse functions? Go to the online text: Inverse functions.

Exponential growth

An exponential growth model has the form

$Q(t) = Q_0e^{kt} \qquad$ (k, Q₀ both positive)

$Q_0$ represents the value of $Q$ at time $t = 0$, and $k$ is the growth constant. The doubling time $t_d$ of a substance undergoing exponential growth is the amount of time it takes for the original quantity to double. The doubling time does not depend on the original quantity $Q_0$ of substance.

The growth constant $k$ and doubling time $t_d$ are related by the equation

$t_dk = \ln 2.$

Exponential decay

An exponential decay model has the form

$Q(t) = Q_0e^{-kt} \qquad$ (k, Q₀ both positive)

$Q_0$ represents the value of $Q$ at time $t = 0$, and $k$ is the decay constant. The half-life $t_d$ of a substance undergoing exponential decay is the amount of time it takes for half the original quantity to decay. As with the doubling time for exponential growth, the half-life does not depend on the original quantity $Q_0$ of substance.

The decay constant $k$ and half-life $t_d$ are related by the equation

$t_hk = \ln 2.$

Logistic functions

A logistic function has the form

$f(x) = \frac{N}{1 + Ab^{-x}}$

for given constants $A$, $N$, and $b$ ($b > 0$ and $b \neq 1$).

Properties of the logistic curve

• The graph is an S-shaped curve sandwiched between the horizontal lines $y = 0$ and $y = N.$ $N$ is called the limiting value of the logistic curve.
• If $b \gt 1$, the graph rises; if $b \lt 1,$ the graph falls.
• The $y$-intercept is $\frac{N}{1 + A}$.
• Role of $b$: For small values of $x$, the function is approximately exponential with base $b$:
  $f(x) \approx \Bigl(\frac{N}{1+A}\Bigr)b^x.$
  .

$b \gt 1$

$b \lt 1$