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## Notes on Logarithmic Computations

### Gaussian Logarithms

Tables of Gaussian Logarithms can be used to calculate the addition or subtraction of number values for which (e.g. due to a certain calculation scheme) only the logarithmic values are available. Using Gaussian Logarithms for this purpose, avoids converting the logarithmic values to the natural number representation and back to the logarithmic value after performing the requested addition or subtraction.

The usefulness of such a table appears first to have been described by Zecchini Leonelli (17xx-18xx) in a book entitled "Théorie des Logarithmes Additionnels et Déductif", published in 1803. The first complete table of Addition- and Subtraction-Logarithms, compiled by Karl Friedrich Gauss (1777-1855), was published in 1812. Gauss' table gives values of log(1+1/x) and log(1+x) tabulated respectively as "B" and "C" as correspondents to argument log(x) tabulated as "A".
Many formulae in spherical trigonometry, which involve addition or subtraction of product term, such as the spherical cosine formula for instance, are unsuitable for direct logarithmic computation. The use of Gaussian logarithms however may overcome this defect.

The Table of Gaussian Logarithms for addition contains the following relationship: the argument of the table is a number log(x) and the corresponding function value is log(1+1/x):

```+-----------+  +----------+----------+
|           |  |    A     |    B     |
|     x     |  |  log(x)  |log(1+1/x)|
+-----------+  +----------+----------+
|     1.000 |  | 0.00000  | 0.30103  |
|     1.002 |  | 0.00100  | 0.30053  |
|     1.005 |  | 0.00200  | 0.30003  |
|     1.007 |  | 0.00300  | 0.29953  |
|     1.009 |  | 0.00400  | 0.29903  |
..         ..         ..
| 50118.723 |  | 4.70000  | 0.00001  |
| 63095.734 |  | 4.80000  | 0.00001  |
| 79432.823 |  | 4.90000  | 0.00001  |
|100000.000 |  | 5.00000  | 0.00000  |
+-----------+  +----------+----------+```

Suppose that log(a) and log(b) are given, with log(b) being smaller than log(a), and that the result for log(a+b) is to be found.

The following relationships hold:

`  A = log(a) - log(b) = log(a/b)`
For the argument A=log(a)-log(b), or x=(a/b), the function value "B" from the Gaussian Logarithms Table is:
`  B = log(1+b/a) = log((a+b)/a)`
If the value of log(a) is added to this result:
`  log(a) + log((a+b)/a) = log(a+b)`
the requested result for log(a+b) is obtained! Notice that the argument x=(a/b) is not explicitly required to obtain this result.

Hence for the logarithm of addition, the following rule is derived:
"Given the numbers a and b for which the logarithmic vales log(a) and log(b) are known and that the logarithmic value of the sum (a+b) is required: log(a+b), then subtract the smaller logarithm from the larger, with the result as an argument take the corresponding Gaussian Logarithm, and add this result to the larger logarithm".

For logarithm of subtraction, the Table of Gaussian Logarithms must be extended with the function C=log(1+x), which is now used as table entry. It is still assumed that a>b so the difference (a-b) is a positive number. Hence if the argument "C" is log(1+x) = log(a) - log(b) or (1+x)=(a/b), the function "B" will give log(1+1/(a/b-1)) = log(a/(a-b)), which subtracted from log(a) will give log(a-b).

```+-----------+  +----------+----------+----------+
|           |  |    A     |    B     |    C     |
|     x     |  |  log(x)  |log(1+1/x)| log(1+x) |
+-----------+  +----------+----------+----------+
|     1.000 |  | 0.00000  | 0.30103  | 0.30103  |
|     1.002 |  | 0.00100  | 0.30053  | 0.30153  |
|     1.005 |  | 0.00200  | 0.30003  | 0.30203  |
|     1.007 |  | 0.00300  | 0.29953  | 0.30253  |
|     1.009 |  | 0.00400  | 0.29903  | 0.30304  |
..         ..         ..         ..
| 50118.723 |  | 4.70000  | 0.00001  | 4.70001  |
| 63095.734 |  | 4.80000  | 0.00001  | 4.80001  |
| 79432.823 |  | 4.90000  | 0.00001  | 4.90001  |
|100000.000 |  | 5.00000  | 0.00000  | 5.00000  |
+-----------+  +----------+----------+----------+```

In case log(a)-log(b) is smaller than log(2)=0.30103, the column "C" cannot be used as table entry. Instead column "B" is used as entry and column "C" as function output. This gives the following relationships: B=log(a)-log(b) or (1+1/x)=(a/b), and thus the function "C" will give log(1+b/(a-b)) = log(a/(a-b)), which subtracted from log(a) will give log(a-b).

Hence for the logarithm of subtraction, the following rule is derived:
"Given the numbers a and b for which the logarithmic vales log(a) and log(b) are known and that the logarithmic value of the difference (a-b) is required: log(a-b), then subtract the smaller logarithm from the larger, with the result as an argument take the corresponding Gaussian Logarithm for subtraction (using the appropriate columns from the table), and subtract this result from the larger logarithm".

#### Examples

Given log(a)=1.2300 and log(b)=0.6700, get the logarithm of the addition: log(a+b). First calculate log(a)-log(b)=0.5600, and enter column "A" with this value to obtain the result from column "B" (with appropriate interpolation): B=0.1056.
Finally, adding "B" to the value of log(a) gives: 1.2300 + 0.1056 = 1.3356, which is the result of log(a+b), with a=16.9824 and b=4.6774 and log(16.9824 + 4.6774) = log(21.6598) = 1.3356!

Using the same numbers for the logarithm of subtraction will require to enter the column "C" with the value log(a)-log(b)=0.5600 and to extract the result from column "B" (again with appropriate interpolation): B=0.1399.
Subtracting "B" from the value of log(a) gives: 1.2300 - 0.1399 = 1.0901, which is the result of log(a-b), with a=16.9824 and b=4.6774 and log(16.9824 - 4.6774) = log(12.3051) = 1.0901!

#### Sources

1. "Logarithmic Computations" by Professor H.A. Howe (University of Denver, Colorado), published in the "Sidereal Messenger", Vol 2, April 1883, pp. 45-50.
2. "Gaussian Logarithms and Navigation" by Charles H. Cotter, in "The Journal of Navigation", Volume 24 - Issue 4, October 1971, pp. 569-572
3. "Gauß'sche Additionslogarithmen feiern dieses Jahr ihren 200. Geburtstag" by Herman Kremer in "de.sci.mathematik", August 2002

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