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(M-13)   Logarithms


    Logarithms can be viewed as a bridge between elementary algebra and more advanced math. As in the rest of "From Stargazers to Starships," here too, differential calculus is avoided, even though it can explain a lot . Therefore these sections may be regarded as an optional addition. After 4 section a natural break occurs--just before the number "e" is introduced, and the user may stop there, or else, continue. And remember that these sections are meant to accommodate a wide variety of users: if any parts seem elementary or repetitious, just fast-forward past them!


    A logarithm of a given number (its "log" for short) is another number, tied to it by some relationship. In other words, it is a "function" of that number. Suppose the chosen number is "2". The user should already know some basic relationships:

    ---The negative opposite (–2). That is the number which, when added to 2, brings one back to zero.

    --The inverse of 2 or 1/2 = 0.5. That is the number which, when multiplied by 2, brings us back to 1, the cornerstone of all multiplication.

    --The square of 2 or 2*2 = 4, the number multiplied by itself.

To these one may add now:

    --The logarithm of 2 or log 2 = 0.301029995... ?
    This number (like √2 = 1.41321..., the square root of 2) cannot be expressed as a fraction (common or decimal) which ends somewhere, but goes on and on. Its defining property is that
2 = 100.3010299.. = 10log 2
That raises several questions:
    --How does one define a power of 2 which are not whole numbers?
    --How does one derive logarithms, i.e. numbers like log2 = 0.3010299..
    --What makes logarithms useful or important?
        You may for instance recall the slide rule used by engineers not long ago, for quick multiplication and division (of limited accuracy, but often good enough for the job). It was based on "logs." How did it work?

An optional example of a calculation using logarithms

To multiply two numbers A an B using logarithms (the pre-calculator way):
    ---Find the logarithms of A and B, using tables.
    ---Add the logarithms
    ---Using tables of "logs", find the number whose logarithm equals the sum.
              (or use a table of "anti-logs" where the value of the logarithm
              is the input and the table then leads you to the number.)
    ---That is your answer.
    The example below shows how it is done--it does not explain why it works (that comes later). Printed tables of logarithms are no longer widely used, but the user of these web pages might instead have an electronic calculator on hand--one which not only adds, subtracts, multiplies and divides, but also has operations like "log" which provide the equivalent of logarithmic tables (computers have their own methods (aka "algorithms") for quickly deriving logariths. (personal computers usually also have that ability). It also helps if it has buttons for operations like 10x or yx; these provide the equivalent of "anti-log" tables, which for a given logarithm tell you what the corresponding number is. In using the yx button, the number y is entered, the button is pushed, then the number x is entered, and "=" gives the result).

    Until about 1970, such calculators were unknown. Mechanical calculators for the first 4 operations--operated by a hand crank, or later by an electric motor--were in limited use from the start of the 20th century, but were expensive, as was their repair. Cash registers used somewhat similar methods, requiring the smooth meshing of many gears, but to multiply two numbers (even large ones), or divide them, most people used paper-and-pencil methods taught in school.

    Those methods can give full precise solutions. However, in many engineering applications, an accuracy of 4 or 5 decimal places (or whatever) is enough. For such applications, tables of logarithms with corresponding precision saved much time and effort, replacing multiplication by an addition--or in other cases, division by subtraction. For reason described below, only logarithms of numbers between 1 and 10 are needed, giving the fractional part of the logarithm, to the right of the decimal point (the "mantissa"). The part to the left of the decimal ("characteristic") encodes the position of the decimal point. Say we wanted (back in the old days) to multiply 123.45 times 98.765 , using logarithms. We would enter

    log 123.45 = 2.091491
    log 98.765 = 1.994603
    Add 4.086094
Now seek in the table of logarithms the number whose logarithm is the fractional part of the sum, in this case 0.086094 (a table of "antilogarithms," often listed with the logarithms, can speed up the search. On your calculator use the button 10x). This gives a number between 1 and 10
The number "4" in front now indicates that the decimal point needs to move 4 steps to the right, so
(123.45).(98.765) = 12,192.53

Actual multiplication gives 12,192.54

To divide 456,789 by 12,345, logarithms are subtracted

    log 456 789 = 5.659716
    log 12 345   = 4.091491
    subtract 1.568224

    The fractional part of the difference is .568224, which equals the logarithm of 3.700197. The "1" in front suggests moving the decimal point one step to the right (the larger the "characteristic", the larger the number being represented!) hence

456,789 / 12,345 = 37.00197

Actual division gives

Here the logarithms were accurate to 6 figures, and the solutions have similar accuracy.

    A third use of logarithms is raising a number to a power (a process expanded to include fractional and negative numbers). This is accomplished by multiplying its logarithm by the index of that power. If one needed


one looked up
log 5.32067 = 0.725966

and multiplied by 3.811 , either using logarithms or the hard way:

(3.811).(0.725966) = 2.766658

The fractional part is
0.766658 = log 5.843298

and the "2" in front tells that the decimal point needs to be moved two steps to the right (making the number larger by two powers of 10), so

(5.32067)3.811   =   584.3298

Does this make sense? Well... 5.32 is close to 5, while 3.811 is close to 4, and these two numbers give 54 = 625, which is "within the ballpark." The yx button on my calculator gives 584.3293, but this agreement is no surprise, since the same calculator was used to derive logarithms and "antilogarithms" (the button 10x) for the above calculation.

    Logarithms thus help complicated mathematical calculations. Nowadays electronic calculators do it all at a push of a button (deriving logarithms in the process and applying them, unseen by the user). Even so, the logarithm of a number often comes up in advanced mathematical calculations and in many applications.

A few quick notes

(1)   The "base" of logarithms

The observant user has probably noted that the definition of the logarithm. e.g.

2 = 100.3010299...   =   10log 2

contains the number "10". Strictly speaking, these are "common logarithms" or "logarithms to base 10," tailored for the convenience of calculations with numbers written in the decimal system. Indeed, purists will indicate the base by a subscript, e.g. 0.3010299... = log102.

    Other "bases" may also be chosen, and one in particular ("natural logs") has advantages of its own. That however will come later, after the number e = 2.71828.... is introduced. Until then, "log" will always indicate a logarithm to base 10.

(2)   Notation for multiplication and parentheses

In what follows below, a period may denote multiplication, either of numbers or of symbols. Following a convention used in practically all technical applications, even that period may be omitted when two clearly different symbols stand side by side:

    2x means 2 times x
    ab           a times b
    abc         a times b times c

However ALWAYS remember--if there exists more than one possible interpretation (or even just to make the operations crystal-clear), use parentheses ( ) or brackets [ ], alerting the user to the correct order. Writing
3.4 + 5
is ambiguous. Most people will multiply first, then add, getting 17, but someone may add first, getting 27. Parentheses make clear which operation comes first:

    (3.4) + 5 = 17
    3.(4 + 5) = 27

Sometimes the order makes no difference, e.g 3.4.5, but it is always safe to write parentheses, even if they are not absolutely required. Parentheses also prevent the multiplication mark from being confused with the decimal point!

(3)   Calculators

    As already noted, deriving logarithms can be tedious business. Until about 1970, students, engineers, technicians and scientists used "log tables" printed in handbooks, giving the logarithms of numbers between 1 and 10 to an accuracy of 4, 5 or 6 decimal places (as in the examples above, logarithms of numbers bigger than 10 or smaller than 1 are simply related to them). Furthermore, they also learned to estimate logarithms of numbers between those tabulated ("to interpolate"), obtaining greater accuracy. You may still find such tables in handbooks.

    In addition, one had tables of "anti-logarithms"--given the logarithm, the table gave the number 10x whose logarithm it was. One could also find the value by searching and interpolating the tables of logarithms, but antilogs saved time.

    Computer chips made all that obsolete, and inexpensive hand-held calculators now can not only add, subtract, multiply and divide, and also find square roots, for which students were taught a moderately tedious routine, a bit more involved than long division. They can also find logarithms and anti-logarithms (in the button function 10x), as well as sines and cosines (for which printed tables were also used, since deriving them is tedious, too). They can even raise fractional powers yx where y and x are any two positive numbers, whole or fractional. (For yx the computer probably first derives some logarithms and then uses them).

    It is assumed here that the user of these web pages has such a calculator, or a computer capable of acting as one.

(4)   History

    Historically logarithms also form sort of bridge between "classical" algebra as known at the time of Galileo (e.g. that of Niccolo Tartaglia, 1500-1557 and Gerolamo Cardano, 1501-76, who developed a solution of the cubic equation and argued over it), and the development of the differential and integral calculus by Newton and Leibnitz in the second half of the 1600s.

   The first form of logarithms was produced in 1614 by John Napier, a Scotts nobleman with an interest in mathematics. His main interest was to produce an aid to calculation, his approach somewhat intuitive. His logarithms were close to the "natural logarithms" defined later, with a slight difference due to his method. He called them "artificial numbers" and published a book about his work (in Latin), which brought him to the attention of Henry Briggs, a professor of mathematics in London.

    Briggs traveled to Scotland in 1615 specifically to meet Napier, and a strong friendship developed between the two. They met again in 1616, but unfortunately, Napier died soon afterwards.

    Briggs and Napier together figured out the system of logarithms with base 10, and with log 1 = 0, as used today. Briggs later derived tables of logarithms (helped by the calculation of square roots) to an accuracy of 8 and later even 14 decimal figures. Kepler was an enthusiastic user of the newly invented tool, because it speeded up many of his calculations, and in 1620 he dedicated one of his publications to Napier.
Quick quiz:

(1) Suppose you have a calculator of the kind described in item (3). You are told that the logarithm of some number is 0.4. How do you find the number itself?

A. Find 100.4 = 2.511886 That is the number you seek, and
log 2.51186 = 0.4

(2) By the same method, you find that the logarithm of 5.62341 is 0.7 Similarly,

    log 56.2341 = 1.7
    log 562.341 = 2.7
    log 5623.41 = 3.7

What do you conclude?

A. You conclude that the fractional part of a logarithm of a number gives you its structure. The whole-number part only tells you where to place the decimal point, as will be later justified.