The Logarithm Function

A brief review of properties

The logarithm function shows up throughout mathematics and computer science. In this note, we will give a brief survey of the logarithm function’s properties.

Definition and Basic Properties

The logarithm function is typically defined as the inverse of the exponential function. That is, given values \(b > 0\) with \(b \neq 1\) and \(x > 0\), \(y = \log_b x\) is the unique solution to the equation \(b^y = x\). Here \(b\) is referred to as the base of the logarithm. Throughout this course, we will us \(\log\) (without the subscript) to refer to \(\log_2\).

The (base-\(2\)) logarithm function has some useful (and curious) properties, listed below.

  1. \(\log 2 = 1\),
  2. \(\log(x \cdot y) = \log(x) + \log(y)\),
  3. \(\log(x^a) = a \log(x)\).
  4. Change of base formula:

    \[\log_b(x) = \frac{\log(x)}{\log(b)}\]

These are the basic properties characterizing the logarithm function, and other useful properties can be derived from these as well. For example,

\[\log(x / y) = \log(x \cdot y^{-1}) = \log(x) + \log(y^{-1}) = \log(x) - \log(y).\]

The second equality comes from property 2, while the third comes from property 3.

Growth of the Logarithm

One characteristic of \(\log n\) is that it grows very slowly with its input. Property 2 above gives some indication of why \(\log n\) grows so slowly: \(\log (2n) = \log (n) + \log (2) = \log(n) + 1\). That is, if we double the input to the \(\log\) function, its value only increases by \(1\). In fact we can show that \(\log n\) grows more slowly than every positive power of \(n\). (The statement below uses big O notation.) We will frequently appeal to the following fact about \(\log n\).

Fact. For every positive number \(a > 0\), \(\log n = O(n^a)\).

To get a more concrete sense of how slowly the logarithm function grows, note the following powers of \(2\):

power of \(2\)   approximate value
\(2^{10}\) \(\approx 1,000\)
\(2^{20}\) \(\approx 1,000,000\)
\(2^{30}\) \(\approx 1,000,000,000\)

Correspondingly, we get following table of logarithms:

logarithm   approximate value
\(\log(1,000)\) \(\approx 10\)
\(\log(1,000,000)\) \(\approx 20\)
\(\log(1,000,000,000)\) \(\approx 30\)

In class, we showed that the binary search procedure for finding an element in a sorted array has a running time of \(O(\log n)\). The table above gives some indication of how efficient binary search is. For example, suppose we use binary search to find a student in an array containing all Amherst College students (\(n \approx 2,000\)), and find the program takes about 10 ms to complete. If we used the same procedure to find a name in a sorted array containing the names of all Massachusetts residents (\(n \approx 7,000,000\)), we might expect that the program would only take about twice as long to complete: 21 ms. Running the same program on a sorted array containing the name of every person on earth (\(n \approx 8,000,000,000\)) would only require about 3 times as many operations as searching for an Amherst College students, so we might expect our program to complete in about 30 ms! Even though the world population is about 4 million times Amherst College’s population, a procedure with \(O(\log n)\) running time might only take 3 times as long to run with the entire world population as its input as the same procedure run on just Amherst College students.