Logarithms: History and Common Bases

Logarithms were originally invented by John Napier (1515-1617) to aid in arithmetical computations at a time when modern day calculators were not in use. In the present day, logarithms have many uses in disciplines as different as economics, computer science, engineering and natural sciences. Logarithmic scales reduce wide-ranging quantities to tiny scopes. For example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and of geometric objects called fractals. They describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting.

While any positive number can be used as the base of a logarithm, not all logarithms are equally useful in practice. Some bases have more applications than others. Out of the infinite number of possible bases, three stand out as particularly useful. These are $10$, $e$ and $2$.

The Common Logarithm

A logarithm with a base of $10$ is called a common logarithm and is denoted simply as $logx$. Common logarithms are often used in physical and natural sciences and engineering. 

The Natural Logarithm

A logarithm with a base of $e$ is called a natural logarithm and is denoted $lnx$. The irrational number  $e\approx 2.718 $ and arises naturally in financial mathematics in computations having to do with compound interest. Natural logarithms are also used in physical sciences and pure math. 

The Binary Logarithm

A logarithm with a base of $2$ is called a binary logarithm and is denoted $ldn$. Binary logarithms are useful in any application that involves the doubling of a quantity, and particularly in computer science with the use of integral parts.

Uses of Logarithms

The list below highlights only some of the many uses of logarithms in the present day.

1. The magnitude of an earthquake (M) can be determined based on the logarithm of an intensity measurement from a seismograph (I):

$M=log(\frac{I}{I_0})$ where I₀ is a constant. 

2. The common logarithm is used in calculating the safety index which helps determine how safe certain activities are by determining how likely people are to die from them. For example, $1$ in $2,000,000$ people is killed by lightning. It might be hard to get a real sense of how likely one is to die by getting hit by lightning because $2,000,000$ is such a large number and our minds cannot make sense of it easily.  If one in $x$ people die as a result of doing some given activity each year, the safety index for that activity is simply the logarithm of $x$. The higher the safety index, the safer the activity in question. The safety index, because it is a logarithm, is a much smaller number. Logarithms help to shrink the numbers of very high magnitude to a smaller ones, which our brains can deal with easily. 

3. pH is an abbreviation for power of hydrogen. The pH scale measures how acidic or basic a substance is. It ranges from $0$ to $14$. A pH of $7$ is neutral (water). A pH less than $7$ is acidic, and a pH greater than $7$ is basic. The acidity depends on the hydrogen ion concentration in the liquid (in moles per liter) written as [H+]. The greater the hydrogen ion concentration, the more acidic the solution. It is defined as $pH = -log10[H+]$. 

Pure water contains a hydrogen ion concentration of $1 \cdot 10^{-7}$moles. It has a pH level of $7$.  Clearly, $7$ is an easier number for our brain to handle. This is an example of how logarithms helps us to deal with numbers of very small magnitudes. 

4. The entropy $(S)$ of a system can be calculated from the natural logarithm of the number of possible microstates $(W)$ the system can adopt:

$S=k \cdot ln(W)$ where $k$ is a constant.

5. Logarithms occur in definitions of the dimension of fractals. Fractals are geometric objects that are self-similar: small parts reproduce, at least roughly, the entire global structure. The Sierpinski triangle can be covered by three copies of itself, each having sides half the original length. This makes the Hausdorff dimension of this structure $ln(3)/ln(2) ≈ 1.58$. Another logarithm-based notion of dimension is obtained by counting the number of boxes needed to cover the fractal in question. 

Sierpinski triangle

The Sierpinski triangle can be covered by three copies of itself, over, and over, and over.

6. Logarithms are related to musical tones and intervals. In equal temperament, the frequency ratio depends only on the interval between two tones, not on the specific frequency, or pitch, of the individual tones. For example, the note A has a frequency of $440$ Hz and B-flat has a frequency of $466$ Hz. The interval between A and B-flat is a semitone, as is the one between B-flat and B (frequency $493$ Hz). Therefore, logarithms can be used to describe the intervals: an interval is measured in semitones by taking the base-$\frac {21}{12}$ logarithm of the frequency ratio.

7. Natural logarithms are closely linked to counting prime numbers ($2, 3, 5, 7$ ...), an important topic in number theory. The prime number theorem states that for large enough N, the probability that a random integer not greater than N is prime is very close to $\frac {1} {log(N)}$.