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Published online by Cambridge University Press: 05 June 2012
Logarithms log2(n) and exponentials 2n arise often when analyzing algorithms.
Uses: These are some of the places that you will see them.
Divide a Logarithmic Number of Times: Many algorithms repeatedly cut the input instance in half. A classic example is binary search (Section 1.4): You take something of size n and you cut it in half, then you cut one of these halves in half, and one of these in half, and so on. Even for a very large initial object, it does not take very long until you get a piece of size below 1. The number of divisions required is about log2(n). Here the base 2 is because you are cutting them in half. If you were to cut them into thirds, then the number of times to cut would be about log3(n).
A Logarithmic Number of Digits: Logarithms are also useful because writing down a given integer value n requires ｢log10(n + 1)｣ decimal digits. For example, suppose that n = 1,000,000 = 106. You would have to divide this number by 10 six times to get to 1. Hence, by definition, log10(n) = 6. This, however, is the number of zeros, not the number of digits. We forgot the leading digit 1.
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