Summary of Rules
(1) The Nth power xN of a number x was originally defined as x multiplied by itself, until there is a total of N identical factors. By means of various generalizations, the definition can be extended for any value of N that is any real number.
(2) The logarithm (to base 10) of any number x is defined as the power N such that
x = 10N
(3) Properties of logarithms:
(a) The logarithm of a product P.Q is the sum of the logarithms of the factors
log (PQ) = log P + log Q
(b) The logarithm of a quotient P / Q is the difference of the logarithms of the factors
log (P / Q) = log P – log Q
(c) The logarithm of a number P raised to power Q is Q.logP
log[PQ] = Q.logP
The usefulness of logarithms comes from the fact that
(a) The first rule allows one to convert a tedious multiplication problem into one requiring only addition.
In all these cases, by expressing the numbers in scientific notation, only logarithms of numbers between 1 and 10 are needed, covering the range from 0 to 1.
(b) The second rule allows one to convert a tedious division problem into one requiring only subtraction.
(c) The third rule allows one to evaluate any power Q of a number P, even if Q is not a whole number.
In order to use those rules we need some method of (a) deriving the logarithm logQ of any given number Q, and (b) doing the opposite--given the value of log Q, find the value of Q.
Up to about 1970, (a) was done by tables of logarithms, giving the result to a certain number of decimals
To go back from the value of the logarithm to its originating number, we look up on the table of logarithms the two entries which flank the value we have, and by subtracting, find by how much they are separated.
Suppose that at the end of some calculation, the logarithm of the answer is 0.45678. On the table of logarithms, we find that the two flanking numbers are 0.45673 and 0.45686. Their separation in units of the last decimal is 13 (86 minus 73) while the number we want to fit is 5 units from the bottom of the interval (78 minus 73). Our number is therefore 5/13 of the length of the interval from its lower end. The two flanking numbers on the table suggest that our answer is between 2.86240 and 2.86325, since
0.45673 = log 2.86240
The difference is 325 – 240 = 85 in the last decimal place, and 5/13 of this is (85)(5/13) ~ 33, so adding 33 to the last digits of the lower number gives an approximation of
0.45686 = log 2.86325
0.45678 ~ log 2.86273
This process, called interpolation, is often useful in finding the value of a function corresponding to a point between two tabulated (or graphed) values.
On today's electronic calculators, simply press the "log" button to get the logarithm and the "10x" button to retrieve the original number (the yx button may also be used). You can try it with the above numbers!
Final things to remember:
---Logarithms are derived for positive numbers only. In a calculation where negative numbers are multiplied or elevated to some power, it may be best to put the minus sign aside for safekeeping, perform the calculation with positive numbers and then figure out (from the nature of the calculation) whether the minus sign should be restored or omitted.
---There is no logarithm of zero. Such a number--call it x--should satisfy 10x = 0, and NO power of 10 equals zero. As an earlier table shows, the diminishing quantities (0.1), (0.01), (0.001), (0.0001) and so forth have logarithms (–1),( –2),( –3),( –4) and so on, becoming more and more negative. Formally one might say the logarithm of zero is at the end of that line and therefore equals minus infinity--except, that "infinity" lacks the usual properties one expects of a number and is therefore not counted as one.
Deriving Approximate Logarithms
So far the only logarithms we have are powers of 10, and they all equal to whole numbers of either sign, or zero (log 1). That is hardly useful!
Using scientific notation
57,140.87 = 5.714087 104
In what follows crude approximations of logarithms will be derived. If you consider going further in studying logarithms, you will be rewarded by the derivation of more accurate values in section M-18.
Once again: applying the Scientific Notation of Numbers
As noted earlier, all one needs is logarithms of numbers from 1 to 10, which go from 0=log 1 to 1=log 10 (all such numbers, not just the integers). With this range covered, numbers outside it can be handled too--essentially, by writing them in scientific notation (or doing something equivalent). Suppose we seek
Since log 104 = 4 , and the logarithm of a product is the sum of the logarithms of the factors
log 57,140.87 = log(5.714087) + log(104) = log(5.714087) + 4
We thus have the logarithm of a number between 1 and 10, which turns to equal to 0.756947 (6-figure accuracy), and adding 4 gives
log 57,140.87 = 4.756947
Approximate Logarithms for Integers from 1 to 10
Below is a list of powers of 2, each twice the one which precedes it:
Index N of power
210 = 1024, but here let it be approximated by 1000 (and by the way, calling 1024 bytes of computer memory "one kilobyte" involves a similar approximation!). So if
210 ~ 1000
taking the logarithms of both sides
10 log 2 ~ 3
log 2 ~ 0.3
A more accurate value is 0.301029995... , so this is not too bad. Three more approximations follow at once:
4 = 22 so log 4 = 2 log 2 ~ 0.6
8 = 23 so log 8 = 3 log 2 ~ 0.9
5 = 10 / 2 so log 5 = log 10 – log 2 ~ 1 – 0.3 = 0.7
Next the powers of 3 are 3, 9, 27, 81 ... so 34 = 81 . Let us approximate it by 80
34 ~ 80 = 8 . 10
take the logs
4 log 3 ~ log 8 + log 10 ~ 0.9 + 1.0 = 1.9
log 3 ~ 0.475
A more accurate value is
log 3 = 0.477121254...
so again... not too bad.
That allows two more logarithms to be approximated
9 = 32 log 9 = 2 log 3 ~ 0.95
6 = 2.3 log 6 = log 2 + log 3 ~ 0.3 + 0.475 ~ 0.775
and finally 74 = 2401 = (3)(8)(100)
Taking logs of both sides and using our approximations
4 log 7 ~ 0.475 + 0.9 + 2 = 3.375
log 7 ~ 0.84375
while more accurately
log 7 = 0.845098...
Again, not too bad
Don't expect accurate results from expressions which are often just one decimal digit deep. For instance, here 3 log5 = 7 log 2, while the numbers they represent are 125 and 128. Still, trying to calculate with these crude logarithms demonstrates the general method.
Further exploration: [for those with access to a university library] "Logarithms!" in American Journal of Physics 46 p. 101, 1978
(1) Suppose we seek (random choice) x = 532 times 373. Using scientific notation
532 = 5.32 102
373 = 3.73 102
(2) Consider finding the logarithm of 5.32, using our approximation: we have the logarithms of 5 and 6, and now need find some intermediate value close to log 5.32 (this process--"interpolation"--was already described. It is the usual method for finding intermediate values on a printed table accurate to, say, 5 decimal figures).
On our table, the difference between log 5 and log 6 is 0.075. If we want to advance 0.32 of that distance, that comes to (0.32).(0.075) = 0.024. So adding 0.024 to our (crude) value log5 = 0.7 gives
Similarly, the difference between log 3 and log 4 on our crude table is 0.125. To get log 3.73 we add to log3 the quantity (0.73).(0.125) or 0.091, so
log 3.73 ~ 0.566 (0.475 + 0.091)
The sum of the logarithms
log [(5.32)(3.73)] = log 5.32 + log 3.73
0.724 + 0.566 = 1.29
(3) Now to find the product (5.32).(3.73) : what is 1.29 logarithm of? Ignore the whole-number part, which will just contribute to the power of 10 in the final result. In scientific notation, the "main part" has logarithm equal to the fractional part 0.29. That is just 1/30 less than 0.3, our approximation to log 2. So we expect that result to be near 2, times some power of 10.
However, since 0.29 is less than log 2 (crude approximation) by 1/30 of its value, we expect the "main part" to be 1/30 less than 2, which gives 1.967.
To this some power of 10 needs to be added. Now the whole part of the logarithm gets
--- 2 from the factor 102 in 532
So by our approximation, if log x = 5.29
---2 more from the factor 102 in 373
---1 more from the part to the left of the decimal after logarithms are added,
--for a total of 5.
x = 1.967 105= 196,700
Multiplying by calculator gives 198,436. The above result is therefore inaccurate, just like its tools; but had our logarithms been accurate to 6 or 7 figures, similar steps would still be followed, and we would have much better accuracy. That might have been the usual method a century ago: today, logarithms are no longer a prime computing tool, but other uses remain.
Note: Just as multiplication and division are opposites, or addition and subtraction, so "taking a logarithm" may sometimes be viewed as opposite of "raising to a power." If you draw a graph of y = 10x and flip it around the diagonal line y=x through the origin, the graph you get resembles the graph (in regular x-y axes) of y = log10 x, the kind of logarithms ("to base 10") we will deal with below.
Writing the Logarithms of Numbers less than 1
As noted, logarithms of numbers with the same digits but different location of the decimal point differ only in the digits giving the whole number. Suppose we use the 5-decimal approximation
log 2 = 0.30103
log 2000 = log 2 103 = 3 + 0.30103 = 3.30103
log 200 = log 2 102 = 2 + 0.30103 = 2.30103
But how about
log 0.2 = log 2 10–1 = –1 + 0.30103 = –0.69897
log 0.002 = log 2 10–3 = –3 + 0.30103 = –2.69897
Here the intuitive link to log 2 is suddenly lost, since the decimal fraction now differs!
Different methods exist to keep that link. Some people write
log 0.2 = –1 + 0.30103
log 0.2 = 0.30103 – 1
and still others write 1.30103 but then put the minus sign above the "1" to show it is a negative quantity (OK for handwriting, but harder to type). Choose whatever you prefer, but be aware that slightly different rules apply to numbers below 1!
The Slide Rule
The slide rule is a calculating device based on logarithms, widely used by engineers before pocket calculators were introduced (some students even wore it in a holster hanging from the belt).
The basic component of the slide rule is the logarithmic ruler. That is a ruler marked with numbers from 0 to 1 (plus finer divisions, typically down to the 3rd decimal figure), but divided unequally, so that the distance of any marked division from the "0" end was proportional to its logarithm.
Two strips with such scales were attached side by side, with one able to slide along the other. Actually, the sliding strip usually rode in a slot in the middle of the rule. A third essential component was the cursor, a transparent slider able to slide up and down the assembly, with a marked line perpendicular to the logarithmic rules. Its role was to mark a position on the ruler.
Suppose we wanted to multiply two numbers A and B, between 1 and 10. Instead of adding their logarithms on paper and then finding the "antilogarithm" of the sum, here we add the lengths of two sections of the scale, then go back to the original scale to find what number the combined length corresponds to.
For instance, suppose we multiply 3 times 2. On the fixed scale, find "3" and place the cursor on its mark. The way the rule is constructed makes the distance of the cursor from the beginning mark proportional to log 3.
Then slide the moving scale until its beginning mark is exactly under the cursor.
Next move the cursor (and slide rules are made so that such motion does not disturb the position of the moving ruler--once it is put in a position, it stays there). Move it over the number 2 on the movable ruler.
The distance of the cursor from the beginning of the movable ruler represents log 2, and its distance from the beginning of the fixed ruler is now (log2 + log3). Check the number on the fixed ruler under the new position of the cursor: that is the product! (And deriving 3.45 times 2.34 is just as quick).
For more instructions, see http://en.wikipedia.org/wiki/Slide_rule
Of course, some operations seem to require placing the cursor beyond the end of the scale. Suppose you multiply 3 times 4: following the above procedure, you will find that "4" on the moving ruler is on the section extending past the end, which the cursor cannot reach. In that case, do not place the beginning of the sliding ruler under the cursor, but its far end. Moving the cursor to cover "4" now, you will find that it makes "1.2" on the fixed scale. Of course, you will have to adjust the decimal point, but again, multiplying 3.98 times 4.32 is just as fast.
That is just the beginning. To divide, you use the slide to subtract lengths, and most rules have several scales, on both sides. The slide rule in the pictures here has divisions 1-10 on one side of the sliding scale and 1-100 (two scales of 1-10, half as large) on the other side. Using the 1-100 scale helps carry out multiplications like 3 times 4 without reversing sides--but since the scale is smaller, accuracy is reduced too.
Also, the division of "9" on the scale 1-100 corresponds to "3" on the scale 1-10. So putting the cursor on x on the 1-10 scale and then looking for the corresponding number on 1-100 gives x2, though the power of 10 in the scientific notation needs to be derived or guessed. Similar, going in the opposite direction can give the square root √x or x0.5. Here, however, caution is needed in applying the scientific notation. Putting the cursor on 3 (on the scale 1-100) helps derive square roots of numbers with even number of decimal figures--3, 300, 30,000, 3,000,000 and so forth, also 0.03. 0.0003 etc. . Putting it on 30 helps derive square roots of numbers with odd number of decimals-
30, 3000, 300,000 and also 0.3, 0.003 etc.
Variations exist--circular scales, for instance. You can still buy slide rules today over the world-wide web, though they are likely to be used ones.
Logarithms in Nature
Only a few examples can be given.
(1) Star magnitudes were first defined by Hipparchus, who charted the heavens in ancient Greece. The brightest stars were assigned magnitude "1", next level was "2", until stars of magnitude 6 were just at the edge of being visible, with good eyesight on a clear night. It was a rather subjective classification.
Today astronomers can measure the light output L of stars electronically, and using this (with allowance for color variation!) they reassigned "stellar magnitude" M to roughly fit the values of Hipparchus. The formula (due to Pogson, 1856) is
M = const – 2.512 log L
The brighter the star, the larger L and the smaller the magnitude (the units in which L is measured do not matter--they only change the constant). It turned out that with this formula, the brightest stars pushed the scale past 1 to fractions and even negative values: Sirius has magnitude –1.5, Jupiter may reach –2.8 and Venus at its brightest is –4.4.
At the other end, time exposures on large telescopes (the "deep field" image of the Hubble space telescope) have mapped magnitudes down to +30.
(2) The Weber-Fechner law, an observational law from the 19th century, states that human senses react logarithmically. If R is what we feel and S is the stimulus
R = const + A log S
The law has been modified since, but is still a rough guide. For example, sound intensity is measured in a unit called bel (in honor of Alexander Graham Bell) and though a sound of 5 bels sounds like just one notch louder than 4 bels, in fact it carries 10 times the energy. And a tenth of a bel is of course a decibel, a term more widely used.
(3) The Richter scale of earthquakes measures the logarithm of the energy released. An earthquake registering 8.2 on the Richter scale carries 10 times more energy than one of 7.2
(4) The velocity attained by a rocket (neglecting staging, etc.) increases only as the logarithm of the mass of fuel. This is demonstrated in rough fashion at the end of the section on rockets ("Rocket Motion"). It is the slow growth of the logarithm which makes high speed launches difficult and expensive!
(5) In chemistry the pH index ("percentage Hydrogen") measures the logarithm of the concentration of hydrogen ions in watery solutions of acids and bases. A value pH=0 is the limit of strong acids, pH=1 has 10 times fewer hydrogen ions but is still a very strong acid, pH=2 is less, and pH=7 is neutral, no acidity or alkalinity. Alkalinity is the concentration of OH negative ions and mirrors acidity: pH=8 is slightly alkaline, pH=9 more so, up to pH=14, the limit for strong acids.
---Given the approximate values found earlier, what are the approximate logarithms of 20, 200, 0.16, 2007 (approximate 223 as 224), 0.125?
---If the constant π (=3.14...) is approximated at the square root of 10 (3.16...), what is the approximate value of its logarithm?
---We seek log 460. If log 4 = 0.6, log 5 = 0.7, what is approximately log 460?
---An earthquake of magnitude 5.6 shook the north of San Jose, California, on Halloween 2007. How many times more energy than magnitude 5? Use our approximate logs.
---Comet Holmes is an old dim comet, reaching just barely past the orbit of Jupiter and known since the 19th century. It was usually pegged at the 17th magnitude, but in October 2007 it suddenly brightened to reach 3rd magnitude (the cause remains to be established). How many times brighter did it become?
(See also here, here, here and here. And did you get a brightening by about 375,000 times?)
Here ends the brief introduction to common logarithms and some of their uses. If all you wanted was a basic understanding, you may stop at this place, though some more advanced aspects are covered in the following sections.