Common logarithm

In mathematics, the common logarithm is the logarithm with base 10.^[1] It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logarithmus decimalis^[2] or logarithmus decadis.^[3] It is indicated by $log(x)$ ,^[4] $log 10 (x)$ ,^[5] or sometimes $Log(x)$ with a capital $L$ ;^[note 1] on calculators, it is printed as "log", but mathematicians usually mean natural logarithm (logarithm with base e ≈ 2.71828) rather than common logarithm when writing "log". To mitigate this ambiguity, the ISO 80000 specification recommends that $log 10 (x)$ should be written $lg(x)$ , and $log e (x)$ should be $ln(x)$ .

Before the early 1970s, handheld electronic calculators were not available, and mechanical calculators capable of multiplication were bulky, expensive and not widely available. Instead, tables of base-10 logarithms were used in science, engineering and navigation—when calculations required greater accuracy than could be achieved with a slide rule. By turning multiplication and division to addition and subtraction, use of logarithms avoided laborious and error-prone paper-and-pencil multiplications and divisions.^[1] Because logarithms were so useful, tables of base-10 logarithms were given in appendices of many textbooks. Mathematical and navigation handbooks included tables of the logarithms of trigonometric functions as well.^[6] For the history of such tables, see log table.

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Transcription

Welcome to the logarithm presentation. Let me write down the word logarithm just because it is another strange and unusual word like hypotenuse and it's good to at least, see it once. Let me get the pen tool working. Logarithm. This is one of my most misspelled words. I went to MIT and actually one of the a cappella groups there, they were called the Logarhythms. Like rhythm, like music. But anyway, I'm digressing. So what is a logarithm? Well, the easiest way to explain what a logarithm is is to have first-- I guess it's just to say it's the inverse of taking the exponent of something. Let me explain. If I said that 2 to the third power-- well, we know that from the exponent modules. 2 the third power, well that's equal to 8. And once again, this is a 2, it's not a z. 2 to the third power is 8, so it actually turns out that log-- and log is short for the word logarithm. Log base 2 of eight is equal to 3. I think when you look at that you're trying to say oh, that's trying to make a little bit of sense. What this says, if I were to ask you what log base 2 of 8 is, this says 2 to the what power is equal to 8? So the answer to a logarithm-- you can say the answer to this logarithm expression, or if you evaluate this logarithm expression, you should get a number that is really the exponent that you would have to raised 2 to to get 8. And once again, that's 3. Let's do a couple more examples and I think you might get it. If I were to say log-- what happened to my pen? log base 4 of 64 is equal to x. Another way of rewriting this exact equation is to say 4 to the x power is equal to 64. Or another way to think about it, 4 to what power is equal to 64? Well, we know that 4 to the third power is 64. So we know that in this case, this equals 3. So log base 4 of 64 is equal to 3. Let me do a bunch of more examples and I think the more examples you see, it'll start to make some sense. Logarithms are a simple idea, but I think they can get confusing because they're the inverse of exponentiation, which is sometimes itself, a confusing concept. So what is log base 10 of let's say, 1,000,000. Put some commas here to make sure. So this equals question mark. Well, all we have to ask ourselves is 10 to what power is equal to 1,000,000. And 10 to any power is actually equal to 1 followed by the power of-- if you say 10 of the fifth power, that's equal to 1 followed by five 0's. So if we have 1 followed by six 0's this is the same thing as 10 to the sixth power. So 10 to the sixth power is equal to 1,000,000. So since 10 to the sixth power is equal to 1,000,000 log base 10 of 1,000,000 is equal to 6. Just remember, this 6 is an exponent that we raise 10 to to get the 1,000,000. I know I'm saying this in a hundred different ways and hopefully, one or two of these million different ways that I'm explaining it actually will make sense. Let's do some more. Actually, I'll do even a slightly confusing one. log base 1/2 of 1/8. Let's say that that equals x. So let's just remind ourselves, that's just like saying 1/2-- whoops. 1/2. That's supposed to be parentheses. To the x power is equal to 1/8. Well, we know that 1/2 to the third power is equal to 1/8. So log base 1/2 of 1/8 is equal to 3. Let me do a bunch of more problems. Actually, let me mix it up a little bit. Let's say that log base x of 27 is equal to 3. What's x? Well, just like what we did before, this says that x to the third power is equal to 27. Or x is equal to the cubed root of 27. And all that means is that there's some number times itself three times that equals 27. And I think at this point you know that that number would be 3. x equals 3. So we could write log base 3 of 27 is equal to 3. Let me think of another example. I'm only doing relatively small numbers because I don't have a calculator with me and I have to do them in my head. So what is log-- let me think about this. What is log base 100 of 1? This is a trick problem. So once again, let's just say that this is equal to question mark. So remember this is log base 100 hundred of 1. So this says 100 to the question mark power is equal to 1. Well, what do we have to raise-- if we have any number and we raise it to what power, when do we get 1? Well, if you remember from the exponent rules, or actually not the exponent rules, from the exponent modules, anything to the 0-th power is equal to 1. So we could say 100 to the 0 power equals 1. So we could say log base 100 hundred of 1 is equal to 0 because 100 to the 0-th power is equal to 1. Let me ask another question. What if I were to ask you log, let's say base 2 of 0? So what is that equal to? Well, what I'm asking you, I'm saying 2-- let's say that equals x. 2 to some power x is equal to 0. So what is x? Well, is there anything that I can raise 2 to the power of to get 0? No. So this is undefined. Undefined or no solution. There's no number that I can raise 2 to the power of and get 0. Similarly if I were to ask you log base 3 of let's say, negative 1. And we're assuming we're dealing with the real numbers, which are most of the numbers that I think at this point you have dealt with. There's nothing I can raise three 3 to the power of to get a negative number, so this is undefined. So as long as you have a positive base here, this number, in order to be defined, has to be greater than-- well, it has to be greater than or equal-- no. It has to be greater than 0. Not equal to. It cannot be 0 and it cannot be negative. Let's do a couple more problems. I think I have another minute and a half. You're already prepared to do the level 1 logarithms module, but let's do a couple of more. What is log base 8-- I'm going to do a slightly tricky one-- of 1/64. Interesting. We know that log base 8 of 64 would equal 2, right? Because 8 squared is equal to 64. But 8 to what power equals 1/64? Well, we learned from the negative exponent module that that is equal to negative 2. If you remember, 8 to the negative 2 power is the same thing as 1/8 to the 2 power. 8 squared, which is equal to 1/64. Interesting. I'll leave this for you to think about. When you take the inverse of whatever you're taking the logarithm of, it turns the answer negative. And we'll do a lot more logarithm problems and explore a lot more of the properties of logarithms in future modules. But I think you're ready at this point to do the level 1 logarithm set of exercises. See you in the next module.

Mantissa and characteristic

An important property of base-10 logarithms, which makes them so useful in calculations, is that the logarithm of numbers greater than 1 that differ by a factor of a power of 10 all have the same fractional part. The fractional part is known as the mantissa.^[note 2] Thus, log tables need only show the fractional part. Tables of common logarithms typically listed the mantissa, to four or five decimal places or more, of each number in a range, e.g. 1000 to 9999.

The integer part, called the characteristic, can be computed by simply counting how many places the decimal point must be moved, so that it is just to the right of the first significant digit. For example, the logarithm of 120 is given by the following calculation:

\log _{10}(120)=\log _{10}\left(10^{2}\times 1.2\right)=2+\log _{10}(1.2)\approx 2+0.07918.

The last number (0.07918)—the fractional part or the mantissa of the common logarithm of 120—can be found in the table shown. The location of the decimal point in 120 tells us that the integer part of the common logarithm of 120, the characteristic, is 2.

Negative logarithms

Positive numbers less than 1 have negative logarithms. For example,

\log _{10}(0.012)=\log _{10}\left(10^{-2}\times 1.2\right)=-2+\log _{10}(1.2)\approx -2+0.07918=-1.92082.

To avoid the need for separate tables to convert positive and negative logarithms back to their original numbers, one can express a negative logarithm as a negative integer characteristic plus a positive mantissa. To facilitate this, a special notation, called bar notation, is used:

\log _{10}(0.012)\approx {\bar {2}}+0.07918=-1.92082.

The bar over the characteristic indicates that it is negative, while the mantissa remains positive. When reading a number in bar notation out loud, the symbol ${\bar {n}}$ is read as "bar $n$ ", so that ${\bar {2}}.07918$ is read as "bar 2 point 07918...". An alternative convention is to express the logarithm modulo 10, in which case

\log _{10}(0.012)\approx 8.07918{\bmod {1}}0,

with the actual value of the result of a calculation determined by knowledge of the reasonable range of the result.^[note 3]

The following example uses the bar notation to calculate 0.012 × 0.85 = 0.0102:

{\begin{array}{rll}{\text{As found above,}}&\log _{10}(0.012)\approx {\bar {2}}.07918\\{\text{Since}}\;\;\log _{10}(0.85)&=\log _{10}\left(10^{-1}\times 8.5\right)=-1+\log _{10}(8.5)&\approx -1+0.92942={\bar {1}}.92942\\\log _{10}(0.012\times 0.85)&=\log _{10}(0.012)+\log _{10}(0.85)&\approx {\bar {2}}.07918+{\bar {1}}.92942\\&=(-2+0.07918)+(-1+0.92942)&=-(2+1)+(0.07918+0.92942)\\&=-3+1.00860&=-2+0.00860\;^{*}\\&\approx \log _{10}\left(10^{-2}\right)+\log _{10}(1.02)&=\log _{10}(0.01\times 1.02)\\&=\log _{10}(0.0102).\end{array}}

* This step makes the mantissa between 0 and 1, so that its antilog (10^mantissa) can be looked up.

The following table shows how the same mantissa can be used for a range of numbers differing by powers of ten:

Common logarithm, characteristic, and mantissa of powers of 10 times a number
Number	Logarithm	Characteristic	Mantissa	Combined form
n = 5 × 10ⁱ	log₁₀(n)	i = floor(log₁₀(n))	log₁₀(n) − i
5 000 000	6.698 970...	6	0.698 970...	6.698 970...
50	1.698 970...	1	0.698 970...	1.698 970...
5	0.698 970...	0	0.698 970...	0.698 970...
0.5	−0.301 029...	−1	0.698 970...	1.698 970...
0.000 005	−5.301 029...	−6	0.698 970...	6.698 970...

Note that the mantissa is common to all of the $5 \times 10 i$ . This holds for any positive real number $x$ because

\log _{10}\left(x\times 10^{i}\right)=\log _{10}(x)+\log _{10}\left(10^{i}\right)=\log _{10}(x)+i.

Since $i$ is a constant, the mantissa comes from $\log _{10}(x)$ , which is constant for given $x$ . This allows a table of logarithms to include only one entry for each mantissa. In the example of $5 \times 10 i$ , 0.698 970 (004 336 018 ...) will be listed once indexed by 5 (or 0.5, or 500, etc.).

Numbers are placed on slide rule scales at distances proportional to the differences between their logarithms. By mechanically adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale, one can quickly determine that

2 \times 3 = 6

History

Common logarithms are sometimes also called "Briggsian logarithms" after Henry Briggs, a 17th century British mathematician. In 1616 and 1617, Briggs visited John Napier at Edinburgh, the inventor of what are now called natural (base-e) logarithms, in order to suggest a change to Napier's logarithms. During these conferences, the alteration proposed by Briggs was agreed upon; and after his return from his second visit, he published the first chiliad of his logarithms.

Because base-10 logarithms were most useful for computations, engineers generally simply wrote " $log(x)$ " when they meant $log 10 (x)$ . Mathematicians, on the other hand, wrote " $log(x)$ " when they meant $log e (x)$ for the natural logarithm. Today, both notations are found. Since hand-held electronic calculators are designed by engineers rather than mathematicians, it became customary that they follow engineers' notation. So the notation, according to which one writes " $ln(x)$ " when the natural logarithm is intended, may have been further popularized by the very invention that made the use of "common logarithms" far less common, electronic calculators.

Numeric value

The numerical value for logarithm to the base 10 can be calculated with the following identities:^[5]

\log _{10}(x)={\frac {\ln(x)}{\ln(10)}}\quad

\quad \log _{10}(x)={\frac {\log _{2}(x)}{\log _{2}(10)}}\quad

\quad \log _{10}(x)={\frac {\log _{B}(x)}{\log _{B}(10)}}\quad

using logarithms of any available base $\,B~.$

as procedures exist for determining the numerical value for logarithm base $e$ (see Natural logarithm § Efficient computation) and logarithm base 2 (see Algorithms for computing binary logarithms).

Derivative

The derivative of a logarithm with a base b is such that

${d \over dx}\log _{b}(x)={1 \over x\ln(b)}$ , so ${d \over dx}\log _{10}(x)={1 \over x\ln(10)}$ .^[7]

Notes

^ The notation $Log$ is ambiguous, as this can also mean the complex natural logarithmic multi-valued function.
^ This use of the word mantissa stems from an older, non-numerical, meaning: a minor addition or supplement, e.g., to a text. Nowadays, the word mantissa is generally used to describe the fractional part of a floating-point number on computers, though the recommended^[by whom?] term is significand.
^ For example, Bessel, F. W. (1825). "Über die Berechnung der geographischen Längen und Breiten aus geodätischen Vermessungen". Astronomische Nachrichten. 331 (8): 852–861. arXiv:0908.1823. Bibcode:1825AN......4..241B. doi:10.1002/asna.18260041601. S2CID 118630614. gives (beginning of section 8) $\log b=6.51335464$ , $\log e=8.9054355$ . From the context, it is understood that $b=10^{6.51335464}$ , the minor radius of the earth ellipsoid in toise (a large number), whereas $e=10^{8.9054355-10}$ , the eccentricity of the earth ellipsoid (a small number).