Bases

As we’ve seen above, any logarithm is always defined relative to a specified base \(b\). These bases can be any positive (real) number not equal to 1. This means the \(b\) can not be negative, or equal to 0, or equal to 1. The base can not be equal to \(1\) because of the definition of logarithms being, as we saw above, the solution for \(x\) of the equation \(y = b^x\). If the base were \(b=1\), then the logarithm would be the solution for \(x\) of \(y = 1^x\), and this not exist because \(1\) raised to the power of any number always gives \(1\). Likewise, if the base were \(b=0\), then we’d be solving for \(x\) in the equation \(y = 0^x\), and this also has no solution because \(0\) raised to the power of any number will also give 0. If \(b\) were a negative value, the logarithm would lead to complex numbers, and as such are not permitted.

Even though the base \(b\) can in fact be any positive real number not equal to 1, certain bases are used more routinely than others. In particular, base \(10\), base \(2\), and base \(e \approx 2.718282\) are widely used. Base \(10\) is commonly used because we count in the decimal number system. Base \(2\) is commonly used in computer science and information theory because of the underlying code used in almost all computers. Base \(e\), on other hand, is used largely for its many convenient mathematical properties. The number \(e\), sometimes known as Euler’s (pronounced oiler’s) number, is an irrational number (meaning that it has an infinite non-repeating decimal expansion), and crops up in mathematics routinely like other (also irrational) numbers like \(\pi \approx 3.141593\) or \(\sqrt{2} \approx 1.414214\). Logarithms to base \(e\) are known as natural logarithms (what’s so natural about natural logarithms? see here), and they are often the default base for logarithms (see below). Moreover, in most application of logarithms to statistics, such as those mentioned above, logarithms to base \(e\) are used.

The domain of logarithms, i.e., the values for which logarithms can be calculated, are the positive (real) numbers. We can not calculate logarithms for negative numbers because there is no solution for \(x\) of the equation \(y = b^x\) when \(y\) is negative. Likewise, there is no solution for \(x\) of \(y = b^x\) when \(y\) is zero.

Algebraic properties of logarithms

Logarithms have some useful algebraic properties, and in fact this is one of the reason why logarithms are widely used. These rules apply regardless of base, so we can drop any explicit reference to the base in the notation. One very important property is \[ \log(y\cdot z) = \log(y) + \log(z) \quad\text{(the log of a product is the sum of logs)}. \] This can be extended indefinitely. For example, \[ \log(u \cdot v \cdot w \cdot \ldots \cdot y \cdot z) = \log(u) + \log(v) + \log(w) \ldots + \ldots \log(y) + \log(z), \] or more generally \[ \log\left( \prod_{i=1}^{n} y_i \right) = \sum_{i=1}^n \log(y_i). \]

This means we can calculate products of very large or very small number easily. For example, if we have a set of \(n\) relatively large numbers, say 100 numbers each of which is on the order of around 1 million, and we were to calculate their product on a computer, this would lead to a number on the order of \(10^{600}\) which is astronomically large, and too large to even represented ordinarily on a computer. However, we could calculate the logarithm of this product by simply summing the logs of the 100 numbers, and this would lead to a number \(\approx 600\) (assuming we use base 10).

A direct consequence of the above rule, given that exponents are repeated multiplication, is \[ \log(y^n) = n \log(y). \] If we put those two rules together we get \[ \log(\tfrac{y}{z}) = \log(y) - \log(z) \quad\text{(the log of a ratio is the difference of logs)}. \] To see why, remember that \(\tfrac{y}{z} = y \cdot \tfrac{1}{z}\) and \(\tfrac{1}{z} = z^{-1}\), and so \[ \log(\tfrac{y}{z}) = \log(y \cdot z^{-1}) = \log(y) + \log(z^{-1}) = \log(y) - \log(z) \]

These rules can also show us how to change the base of a logarithm. Let’s say you have the logarithm to base \(2\) of a certain number but wanted wanted to know its logarithm to base \(10\). If the logarithm to base \(2\) was \(x\), obviously, we could do \(\log_{10}(2^x)\), but sometimes this is not be computationally feasible (because \(2^x\) might be too large to ordinarily represent on a computer) or might not be convenient. Using one of the above rules, we see that \[ \log_{10}(2^x) = x \log_{10}(2), \] and in general, if \(x\) is the logarithm to base \(a\) and you want its logarithm to base \(b\), do \[ x \log_{b}(a), \] which will always give you \(\log_b(a^x)\).

All of these logarithm rules are themselves just a consequence of the rules of exponents, such as \[ \begin{aligned} b^n \cdot b^m &= b^{(n + m)},\\ \tfrac{b^n}{b^m} &= b^{(n - m)},\\ (b\cdot c)^n &= b^n \cdot c^n,\\ (b^n)^m &= b^{(n \cdot m)},\\ b^{-n} &= \tfrac{1}{b^n}. \end{aligned} \]

Calculating logarithms using R

Calculating logarithms using R is easy. For example, to calculate logarithms to base 2, we would use log2:

x <- c(1.25, 2.75, 15.1, 27, 51)
log2(x) # logarithm to base 2

## [1] 0.3219281 1.4594316 3.9164766 4.7548875 5.6724253

To calculate logarithms to base 10, we use log10:

log10(x) # logarithm to base 10

## [1] 0.09691001 0.43933269 1.17897695 1.43136376 1.70757018

To calculate logarithms to base \(e\), we use log:

log(x) # logarithm to base e

## [1] 0.2231436 1.0116009 2.7146947 3.2958369 3.9318256

In fact, we can use log for calculating logarithms to any permissible base, it just that with log base \(e\) is the default. So, for example, here is the logarithm to base \(1.2345\)

log(x, base=1.2345) # logarithm to base 1.2345

## [1]  1.059229  4.801918 12.886248 15.644843 18.663786

Exponents in R are usually obtained by the ^ operator. For example,

c(2^3, 3^3, 10^2)

## [1]   8  27 100

To obtain \(e^x\), however, we use the exp function. For example, \(e^{10}\) is calculated by

exp(10)

## [1] 22026.47

And we could always obtain the value of \(e\) itself, by

exp(1)

## [1] 2.718282

Or, to see more digits of \(e\), you can do

format(exp(1), digits = 22)

## [1] "2.718281828459045090796"