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Overview Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Appendix Online Material

Dead Reckoning: Calculating Without Instruments

Chapter 4: Logarithms and Their Inverses

Background and Topics

After a brief review of logarithms, this chapter discusses general approximation techniques for common and natural logarithms that are suitable for mental calculation, then details more accurate relations involving neighboring and intermediate values, offers methods for the more difficult problem of finding inverse logarithms, and concludes with an example of using logarithms and inverses to find higher-order roots of numbers.

Notes and Errata (If you have any more to contribute, please email me! This is a compendium of feedback)  A printer-friendly summary table for all chapters is found here.

Color Code
A clarification or elaboration of the text.
A simple mistake that does not affect the method presented.
An error that may affect a method or the reader interpretation of it.
It is my preference in this book when expressing a decimal value that continues on past the digits that are displayed, that the last digit shown is rounded off, followed by ellipses (“…”). For example, a value of log 2 = 0.3010299956…when shown to 5 decimal places will be given as 0.30103… rather than 0.30103 or 0.30102… This helps a great deal when comparing an approximation to an exact value to a certain number of places.
To be clear, the value of x in the equation halfway down the page would be N.
In Equation 14, the range should read -1 < x <= 1.
In the last equation on the page for log(1+x), the range should read 10^(-m) < x < 10^(-m+1).
2/2423 should read 2/2421 since N=1210. This is not sufficient to affect the result to the number of digits taken.
It may seem that 2 log 11 + log 10 = 3.08279 should be 3.08278 if we use the value given to five decimal places for log 11 on page 119 (1.04139). However, the value of log 11 to 6 decimal places is 1.041393…, so 2 log 11 actually rounds to 2.08279. The purpose on page 122 was to find the ultimate attainable precision of the formula, so the number of digits used for log 11 was not limited.
In Equation 16, and again on page 123, the term 3(2N+1) should be squared. The term 3(2N+a) should be also be squared on page 123.
In the general equation for ln(N+a), the 2a/(2N+a) should be added, not subtracted, and the numerators of the last two terms should be a^2 and a^4, respectively.
In the top equation on this page, the signs in front of the each “½” on the right-hand side of the equation should be reversed.
In the top equation, only one side should have a negative sign.
In the second equation from the bottom on page 126 and the second equation from the top on page 127, the term 2d should be simply d. This does not affect the validity of the ultimate Equation 18.
“if we actually knew log 22 and log 23” should read “if we actually knew log 22 and log 24”.
The final result 1.1139589 is shown to more digits than the intermediate results in order to show more clearly the accuracy of the formula.
Just to avoid possible confusion, the values a and g are not exactly the same as the arithmetic-geometric mean. The equations show the correct meanings of these.
The equation N'=10^p2^mN should read N=10^p2^mN'.  There is no effect on any results.
In the equation for d3, the numerator is reversed. The result is correct.
In Table 8 the error for 4/3 should be 2x10^(-4).
In the two lines above the result 47.00, a factor of 11/10 is not shown. The result is correct.
“b<=.005” should read “|b|<=.005”.
137-142 Note Tips to lessen the effort when using the Bemer method for antilogarithms/exponentials can be found in the paper The Practical Use of the Bemer Method for Exponentials described and linked in the Additional Materials section below.
Note John McIntosh presents an alternative to the Bemer method (the McIntosh-Doerfler method) for mentally calculating exponentials in his essay Exponentials located here.
Note New methods of approximating exponentials and logarithms can be found in the paper Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions described and linked in the Additional Materials section below.


Additional Materials Related to Topics in This Chapter

The Practical Use of the Bemer Method for Exponentials:  A method is given in Chapter 4 for mentally calculating exponentials, i.e., raising a number to a power where neither the number or power is typically an integer. An example of this is calculating 10^2.85251 = 712.049… This operation 10^n is often called an antilogarithm, which is the terminology used in the book. This paper provides some additional tips that in practice can considerably lessen the effort when using the Bemer method in the book to mentally calculate exponentials. The complete algorithm is presented here; there is no need to have read the book.

Fast Approximation of the Tangent, Hyperbolic Tangent, Exponential and Logarithmic Functions: Several methods of calculating exponential and logarthmic functions are provided in the book, and the most promising exponential algorithm is also described in greater detail in the paper above.  They are very accurate methods (to at least five significant digits), but they can require a certain level of effort.  This paper derives new methods of mine that provide faster calculations of these functions to a respectable accuracy of nearly four-digits.  (It is also my first paper formatted using the LaTeX mathematical typesetting language.)

Cited Reference Materials

A Subroutine Method for Calculating Logarithms:  R.W. Bemer's original article on a method of approximating logarithms that is described in this chapter can be found here.

Logarithms of Large Numbers:  C.C. Camp's 1928 article describing the furious race between nations to calculate precise logarithm tables is interesting for two reasons:  first, it describes some interesting techniques, and second, it is eye-opening to see what mathematicians and their aides had to do by hand before computers were invented.

In the near future, this section will contain additional relevant sections of some of the references cited at the end of this chapter.

Acknowledgements of Contributors to this Page

Once a gain, a big thanks goes to John McIntosh for providing a number of the errata and notes you see on this page.  John also presents a somewhat different method of mentally calculating exponentials here, referenced as well in my paper above on the practical use of the Bemer method.

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