Logarithm tables

In the 17th century, the appearance of logarithms allowed astronomers and navigators to perform more efficient calculations in plane and spherical trigonometry. They especially needed tables of logarithm of sine log(sin(x)) values. This is why the first tables were designed for this purpose.

Of course, all calculations in the 17th, 18th, and 19th centuries were done by hand (paper, pencil). And even at the beginning of the 20th century, with no electronic calculators or computers, at best, mechanical calculators capable only of addition and subtraction with a maximum of 10 digits were available.

Technicians, high school and university students in the 60s/70s often had only logarithm tables, paper/pencils, and a slide rule. The latter, based on logarithmic scales, allowed products and quotients to be performed with an average accuracy of 0.2%. Logarithm tables, on the other hand, allowed precisions of 10^-7 to be reached.

Let's talk about the precision of tables. This is expressed by the number of significant digits in the output (also referred to as the number of decimal places or the number of places). For example, the Bouvart & Ratinet tables intended for high school and university students at the beginning of the 20th century allowed a logarithm with 5 decimal places (significant digits) to be obtained from a number with 4 significant digits (see 6 decimal places by interpolation). The larger the number of places in the inputs, the more voluminous the work.

Attention! In what follows, we will only deal with base 10 logarithms (also called decimal logarithms). Thus, \( \boxed{\log \equiv \mbox{Log}_{10} \equiv \log_{10}} \)

Example:
Let \(x = 3245\), a number with 4 significant digits.
To calculate \(\log_{10}(3245)\), let \(x=3.245 \times 10^3 \), then \(\log_{10} x = 3 + \log_{10}(3.245)\approx 3 + 0.51121 \).

The B&R Table gives

Slide rules have an average accuracy of 0.2% (\(2.10^{-3}\)). Let's see how to obtain a more precise result using logarithm tables.

Some reminders:
To use "Logarithm Tables", the number N whose logarithm is sought must first be rewritten as follows: This also applies to performing operations using a slide rule.
So \(\log(N)= \log(10^c)+\log(x)=c+\log(x)\)
2 examples of rewriting (I use the term normalization), A \(\log\) can be broken down into 2 parts:

• its integer part called characteristic \(c\)
• its decimal part called mantissa \(m=\log(x)\)

$$ \left . \underbrace{ 1 }_{characteristic\ c} \overbrace{.511307037449982...}^{mantissa\ m} \right . $$

The "Bouvart & Ratinet" logarithm tables (denoted B&R Logarithm Table) give a value of logarithms for numbers N having up to 4 significant digits.

It is possible to go up to 5 digits in the input by applying a linear interpolation between 2 bounding values. The precision of the mantissa is 5 significant digits.

There are (voluminous) tables admitting 5 (6 by interpolating) significant digits and 7 digits for the mantissa.

Let's calculate the logarithm of 32.4569

We start by normalizing 32.4569

\(32.4569 = 3.24569 \times 10^1 \Longrightarrow \log(32.4569)=1+\log(3.24569)\). Its characteristic \(c\) will therefore be 1. It remains to find its mantissa \(m=\log(3.24569)\) .

Our table only admits 4 significant digits. We can therefore set \( 3.245 < 3.24569 < 3.246\)

Let's look in our (excerpt from) B&R Logarithm Tables for the entries 3245 and 3246 without worrying about the decimal point.

• For 3.245, we find 51121. This means that \(\log(3.245)\approx 0.51121\)
• For 3.246, we find 51135. This means that \(\log(3.246)\approx0.51135\)

Let's put these values in a table to make it clearer.

Note that the difference between 51135 and 51121 is 14. In our (excerpt from) table, we observe a small table in the upper left numbered 14.

Let's go back to N = 32.45(69) and isolate the digits beyond 3245. That is 69; formed by 6 and 9.

In table [14],

The sum 9.66 x 10^-5 is the linear interpolation correction to add to log(3.245).

Let's compare this result with those given by a calculator and a slide rule

We are far from the poor result obtained with a slide rule

Let's see what a linear interpolation gives to go from \(\log(3.245)\) to \(\log(3.24569)\)

In fact, this method, and therefore multiplications/divisions, can be avoided thanks to the small tables placed in the left or right margins. Only additions/subtractions are necessary.

Excerpt from the B&R Logarithm Tables (page 8)

Logarithm tables also contain logs of sines and cosines. However, the notation used may be surprising. Let's take an example.

Determine log(sin 30°)

\(\sin 30° =0.5 <1 \Longrightarrow\) its logarithm will therefore be negative.

In the B&R table on page 12, we look for the entry 5000 (for 0.5000). Opposite, we can read \(69 897\). So \(\log(\sin 30°) \approx -1 + 0.69897 \approx -0.30103 \).

Now let's go to page 170 which directly gives \(\log(\sin 30°)\). We find \(\log(\sin 30°) \approx \overline{1}.69897\). This strange notation allows negative log values to be written compactly. To summarize,

(in Bouvart & Ratinet tables)

In other tables, Anglo-Saxon or those dating from before the 20th century, another notation is found for negative logarithm values.

(in Anglo-Saxon tables)

\(\sin(30°)=0.5 <1 \Longrightarrow\) its logarithm will therefore be negative. In the B&R table, we look for the entry 5000 (for 0.5000 on page 12). We can read \(69 897\). So \(\log(\sin(30°)) \approx -1 + 0.69897 \approx -0.30103 \).

Now let's go to page 170 which directly gives \(\log(\sin(30°))\). We find \(\log(\sin(30°)) \approx \overline{1}.69897\). This strange notation allows negative log values to be written.

To summarize, \( \boxed{\log(\sin(30°)) \approx \overline{1}.69897 \approx -1 + 0.69897 \approx -0.30103}\)