Logarithms

An attempt to develop intuitions around an unintuitive idea

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The Exponential Function

We start with the exponential function.

The exponent tells us how many times to multiply the base by itself to get the result.

The Logarithm

We then consider the logarithm.

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In words:

For example:

The logarithm answers the question: What is the power to which we have to raise the number to get the result?

The Natural Logarithm

For deep mathematical reasons, it is often useful to use logarithms with base \(e\)¹ which is often termed the natural logarithm, written \(\ln\)

\(e\) emerges in many contexts, including some aspects of compound growth processes². \(e\) is approximately equal to 2.71828….

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Logarithmic Spiral

An interesting sidenote is that the logarithm forms the basis of the logarithmic spiral. The equation for a logarithmic spiral in polar coordinates is: \(r = ae^{b \theta}\), where \(\theta\) is the angle, \(r\) is the radius, and \(a\) and \(b\) are constants.

Logarithmic spirals can be found in nature in the nautilus shell, and in sunflowers and in the flight of hawks.

\[\color{purple}{y} \color{black}{=} \color{blue}{\text{base}}^{\color{red}{\text{exponent}}}\]

\(\color{blue}{2}^{\color{red}{3}} \color{black}{=} \color{purple}8\)

because

\(\color{blue}{2} \times \color{blue}{2} \times \color{blue}{2} \color{black}{=} \color{purple}{8}\)

If…

\[\color{purple}{y} \color{black}{=} \color{blue}{b}^{\color{red}{x}}\]

Then… \[\log_{\color{blue}{b}}{\color{purple}{y}} \color{black}{=} \color{red}{x}\]

If \(\color{purple}{\text{number}} = \color{blue}{\text{base}}^{\color{red}{\text{exponent}}}\)

then

\(\log_{\color{blue}{\text{base}}}{\color{purple}{\text{number}}} = \color{red}{\text{exponent}}\)

\(\color{blue}{2}^{\color{red}{3}} = \color{purple}{8}\)

therefore

\(log_{\color{blue}{2}} \color{purple}{8} = \color{red}{3}\)

TipKey Idea

The logarithm may thus be thought of as the inverse of the exponential function.

Tipe

\(e\) is a kind of fundamental mathematical constant, like \(\pi\).

If

\[\color{purple}{y} = \color{blue}{e}^{\color{red}{x}}\]

Then

\[\ln{\color{purple}{y}} = \color{red}{x}\]

Nautilus Shell, Courtesy Wikipedia

Sunflower, Courtesy Wikipedia

Spiral Flight of Hawk from Livio (2002)

References

Livio, Mario. 2002. The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number. In The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number, 1st ed. Broadway Books.

Footnotes

1. \(e\) does not have the easy geometric definition that \(\pi\) has.

   For any \(\bigcirc\), \[\pi = \frac{\text{circumference}}{\text{diameter}}\].

2. A common definition of \(e\) is \[e = \lim_{n \to \infty} \left(1 + \frac{1}{n} \right)^n\]