# Mathematical Grammar - “laws” or “properties”

I want to ask about the grammar used in mathematics regarding the use of the words laws and properties.

I was reading back on logarithms and manipulating logarithms in calculations.

The following list are the logarithms I am concerning the question about:

$$log_axy = log_ax + log_ay$$ $$log_a\frac{x}{y} = log_ax - log_ay$$ $$log_ax^n = nlog_ax$$ $$log_a1 = 0$$ $$log_aa = 1$$

Now, when studying these, my professor strictly told me the following -

Logarithms have properties, not laws. A law implies that a mathematican has declared e.g. that the addition of two logarithms to be $log_ax + log_ay = log_axy$ whereas a property implies that a mathematician has discovered logarithms to exhibit such behaviour.

Although I am not a mathematically advanced student, I wanted to ask what do people think regarding this terminology, because my lecture delved into a very brief argument regarding this and I thought it would be interesting for me to hear the opinions of users on StackExchange.

Is the above best referred to as logarithm properties or logarithm laws?

• I would call it "laws" because they mention how to use the logairthms. Another word coming into my mind, is "rules". A property would ,for example, be that $\log(x)$ is only defined for $x>0$ – Peter Sep 21 '17 at 17:14
• I would go with properties of logarithms, as a reflection of properties of powers or exponentiation – Henry Sep 21 '17 at 17:17
• see this is where it gets interesting because opinions vary and I can't seem to see whether one argument outweighs the other - which was interesting in my lecture today. – vik1245 Sep 21 '17 at 17:22
• I'd not thought about the distinction until you asked this question. Now that I have I think your professor is absolutely right. These are indeed often called "laws" but ought not be, for exactly the reason your professor gives. Thank him or her from me please. By the way, some of the properties you state are not quite right as written. Check them and edit. – Ethan Bolker Sep 21 '17 at 17:22
• Some people would say (for natural or real numbers) that $a+b=b+a$ is the commutative law. I would see this as a property (or in some cases an axiom). Contrast it with the superficially similar statements $ab=ba$ and $a^b=b^a$, the first of which is true while the second is usually false, but it seems strange to describe these as laws. – Henry Sep 21 '17 at 17:28

I’m with your teacher. We have to emphasize that mathematics deals overwhelmingly in phenomena, not rules that are imposed on things.

Distributivity, according to which $a(b+c)=ab+ac$ for numbers $a$, $b$, and $c$, is not something imposed on numbers (much less on students!) by some officious mathematician, but an observation about how numbers, addition, and multiplication behave.

In the same way, the formula $\log(xy)=\log x+\log y$ is a description of phenomena, and so most appropriately should be called a Property.

Consider the "laws" for abelian additive groups: $$a+b=b+a,\; a+(b+c)=(a+b)+c,\;a+(-a)=0,\;a+0=0+a=a.$$ Any particular abelian additive group, in order to be called an abelian additive group will have to satisfy these laws. In other words, it will have certain properties that include the laws. So, in the abstract, there are laws for groups and any particular group will have properties including satisfying the group laws.

The logarithm 'laws" essentially give the laws for a homomorphism from an abelian multiplicative group to an abelian additive group. The particular function $x\mapsto \log_a(x)$ on $\mathbb{R}^+\to\mathbb{R}$ has properties that satisfies those laws. Thus, depending on context, one can use logrithm "laws" as abstract laws for certain homomorphisms, and as properties for a particular function.

It's a matter of history and tradition. It happened that the laws of indices and logarithms were so named at a time when such a term was in fashion. Then, there was less understanding of the distinction between mathematics and physics. Later, while physics kept the term law, mathematicians tended to prefer other terms, such as property. However, law was not abandoned completely: for example, the law of the iterated logarithm was discovered only well into the 20th century. Perhaps its name was influenced by familiarity with "the laws of logarithms".

In my view, the use of the traditional term is harmless. No-one is going to draw from it the belief that, unlike the rest of mathematics, the behaviour of logarithms is decreed by a diktat of mathematical authorities.

To say that the laws of exponents or logarithms are not laws because they were not legislated makes as much sense as to say that they are not properties because you can't own them, can't buy or sell them, can't build houses on them or bequeath them to your heirs, don't have to pay taxes on them, etc. A word can have more than one meaning. From the long entry for law in the online OED, under the subheading "III. Scientific and philosophical uses", here is an apt quotation from 1865:

A Law expresses an invariable order of phenomena or facts.

Thus the law of gravity, the laws of thermodynamics, the law of sines, the law of the mean, De Morgan's laws, etc.