Assumed knowledge

The content of the modules:

Furthermore, knowledge of the index laws and logarithm laws is assumed. These are covered in the TIMES module Indices and logarithms (Years 9–10) and briefly revised at the beginning of this module.

Motivation

— Albert A. Bartlett

Our world involves phenomena and objects on many different scales.

Repeated multiplication by 10 can rapidly transform a microscopically small number to an astronomically large one. Multiplying by 10 a few times takes us immediately from the scale of atoms and molecules to the scale of microbiology, insects, humans, cities, continents, planets and beyond — from scales that are imperceptibly small to scales that are almost unfathomably vast. There are only 17 orders of magnitude between the size of a single human cell and the size of our solar system. 1 Understanding the functions involved in such repeated multiplication — namely, exponential functions such as \(10^x\) — is a useful step towards a grasp of these enormities.

Exponential functions, with all their properties of sudden growth and decay, arise in many natural phenomena, from the growth of living cells to the expansion of animal populations, to economic development, to radioactive decay. The quote from Professor Bartlett at the start of this section was made in the context of human population growth. The inverses of exponential functions — namely, logarithmic functions — occur prominently in fields as diverse as acoustics and seismology.

To understand these natural processes of growth and decay, it is important, then, to understand the properties of exponential and logarithmic functions.

In this module, we consider exponential and logarithmic functions from a pure mathematical perspective. We will introduce the function \(y=e^x\), which is a solution of the differential equation \(\dfrac{dy}{dx} = y\). It is a function whose derivative is itself. In the module Growth and decay, we will consider further applications and examples.

The module Indices and logarithms (Years 9–10) covered many properties of exponential and logarithmic functions, including the index and logarithm laws. Now, having more knowledge, we can build upon what we have learned, and investigate exponential and logarithmic functions in terms of their rates of change, antiderivatives, graphs and more.

In particular, we can ask questions like: How fast does an exponential function grow? It grows rapidly! But, with calculus, we can give a more precise answer.

A brief refresher

To jog your memory, we recall some basic definitions and rules for manipulating exponentials and logarithms. For further details, we refer to the module Indices and logarithms (Years 9–10).

Logarithms and exponentials are inverse operations. In particular, for \(a>1\),

\[ x = a^y \ \iff \ y = \log_a x. \]

The following index laws hold for any bases \(a,b>0\) and any real numbers \(m\) and \(n\):

\begin{alignat*}{2} a^m a^n &= a^{m+n} &\qquad\qquad\qquad \dfrac{a^m}{a^n} &= a^{m-n}\\ (a^m)^n &= a^{mn} & (ab)^m &= a^m b^m\\ \Bigl(\dfrac{a}{b}\Bigr)^m &= \dfrac{a^m}{b^m}. \end{alignat*}

Some simple consequences of the index laws are, for \(a>0\) and a positive integer \(n\):

\[ a^0 = 1 \qquad\qquad\qquad a^{\frac{1}{n}} = \sqrt[n]{a} \qquad\qquad\qquad a^{-1} = \dfrac{1}{a}. \]

The following logarithm laws hold for any base \(a>1\), any positive \(x\) and \(y\), and any real number \(n\):

\begin{alignat*}{2} \log_a 1 &= 0 &\qquad\qquad\qquad \log_a a &= 1\\ \log_a (xy) &= \log_a x + \log_a y & \log_a \dfrac{x}{y} &= \log_a x - \log_a y\\ \log_a \dfrac{1}{x} &= -\log_a x & \log_a (x^n) &= n \, \log_a x. \end{alignat*}

Also, recall the change of base formula:

\[ \log_b x = \dfrac{ \log_a x }{ \log_a b }, \] for any \(a, b > 1\) and any positive \(x\).

Two approaches

In this module, we will introduce two new functions \(e^x\) and \(\log_e x\). We will do this in two different ways.

The first approach develops the topic in an investigatory fashion, starting from the question: `What is the derivative of \(2^x\)?' However, as we proceed, we will point out some shortcomings of this approach.

Alternatively, we can begin from a definition of \(\log_e x\) as an integral, and then define \(e^x\) as its inverse. The story is then told in a completely different order.

The first approach is probably easier for most students to understand, but the second approach is more concise and rigorous.

In general, when telling a mathematical story, there are various goals such as elegance, rigour, practicality, generality and understandability. Sometimes these goals conflict, and we have to compromise. Sometimes developing a subject in the most logically concise way does not make for easy reading. As with any other subject, learning mathematics from multiple perspectives leads to a deeper and more critical understanding.

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