TAOCP 4.2.4 Exercise 4
An antilogarithm table is indexed by values of $\log_{10} x$.
Section 4.2.4: Distribution of Floating Point Numbers
Exercise 4. [M18] The text points out that the front pages of a well-used table of logarithms get dirtier than the back pages do. What if we had an antilogarithm table instead, namely a table that tells us the value of $x$ when $\log_{10} x$ is given: which pages would get dirtier?
Verified: yes
Solve time: 1m42s
Solution
An antilogarithm table is indexed by values of $\log_{10} x$. To determine which pages are used most frequently, we must determine the distribution of $\log_{10} x$ when the numbers $x$ obey the logarithmic law described in the text.
Write
$$ x = 10^m f_x, $$
where $1 \le 10f_x < 10$. Then
$$ \log_{10} x = m + (\log_{10} x)\bmod 1. $$
The fractional part $(\log_{10} x)\bmod 1$ is uniformly distributed on $[0,1)$, by the argument leading to equation (1). Hence the mantissas of logarithms are uniformly distributed.
An antilogarithm table is consulted by entering with the mantissa of $\log_{10} x$. Since these mantissas occur uniformly, every page of the table is used with essentially the same frequency.
Therefore no pages become noticeably dirtier than others. The wear should be approximately uniform throughout the table.
$$ \boxed{\text{All pages of an antilogarithm table become approximately equally dirty.}} $$