812
The Logarithm Function
Common logarithms were used as computational aids before calculators and
computers became commonplace. Property 7 of the logarithm function says that
to multiply two numbers, one can take the logarithm of each number, add the results, and then find the number with that logarithm value. Addition was easier than
multiplication, and tables of common logarithms allowed one to look up a number
and find its logarithm, or vice versa.
Natural logarithms are useful in calculus and are often written “ln p” rather than
“log e p.” Base 2 logarithms are sometimes denoted by “lg p” rather than “log 2 p.”
In this book, all logarithms are base 2, so we use log p to denote log 2 p.
A final inequality involving base 2 logarithms (used in Section 3.3) is
1 + log n < n for n ≥ 3
To prove this, note that
n < 2
n−1
for n ≥ 3
so by property 1 of logarithms,
log n < log2
n−1
By property 5 of logarithms, log2
n−1
= n − 1. Therefore
log n < n − 1
or
1 + log n < n for n ≥ 3