I like the number 1. It seems a dull choice and many may go for other more mysterious numbers like 4, 8 or 24. But there are some key elementary concepts that hinge on 1.
One of these concepts is the basic operations of multiplication and of division, and in particular the integer division. The basic idea may be stated as what are the “paths” that lead to 1.
For instance, be 1 < b < n integers. The number of times one can divide n by b without the result being smaller than 1 is the integer logarithm of n in base b. Let’s denote it by iLog_b(n).
iLog_3(81) = 4; iLog_2(128) = 7.
For 0 < b < 1, the integer logarithm means the number of times we can multiply n by b without the result being smaller than <1. So we’d write iLog[1/2](8)=3. However, it will turn out to be convenient to introduce a new notation for meaning multiplication by the base: We will convene in adding a negative sign in front of the result when meaning repeated multiplication. Hence
iLog_[1/2](8) = -3
For 1 < b < n, iLog_b(1/n) means the number of times we can multiply 1/n by without the result getting larger than 1. By the same rule as before, we will add a negative sign to denote it’s a multiplication. In addition, it clear is iLog_b(1/n) = – iLog_b(n). Whence,
iLog_[1/2](1/8) = – iLog_2(1/8) = iLog_2(8) = 3
Figure 1 above illustrates the rule of changing base in the logarithm. It also gives us another interpretation of the logarithm: We can read the term Log_[b’](b) as how many cuts (divisions) do we need of size b’ for each cut of size b.
Clearly, iLog_b(1) = 0, be it b>1 or b<1as neither dividing nor multiplying by b will lead as to 1. For for b=1 things are slightly different: we can divide any an infinite number of times by 1 and it still won’t get us below 1, thus iLog_1(n) = ∞ (division). Analogously for multiplication. Following our convention, though, it would be iLog_1(n) = -∞ (multiplication).
If n<0 the logaritm doesn’t exist, as the results of repeatedly dividing/multiplying by b will all be negative and thus never possibly be +1!
Analogously, the logarithm doesn’t exist for b=0 as division by 0 is not defined and 0 is a fixed point for the multiplication.
If after dividing n by b iLog_b(n) times we get exactly 1, we say the logaritm is exact.
If 1 < n < b, we use the symmetry of the logaritm: Log_b(n)*Log_n(b)=1! This follows immediately from the rule of base change by setting b’=x. Of course, for iLog_b(n) it would be zero according to the original definition. Here we will interpret this case as iLog_b(n) = 1 / iLog_[n](b) if 1 < n < b.Hence
Log_(4) * Log_(8) = 1
Log_(4) = 1/Log_(8) .
Let’s prove this reflection symmetry algebraically. In Figure 1 it is illustrated geometrically (The current prove has thus the drawback that relies “too closely” on the idea of power, which is what we are trying to avoid here). For that we will use the rule for a logarithm of a product that we will prove afterwards: Log_b(n*m) = Log_b(n) + Log_b(m), which we assume it holds in the general case -not just for the integer logarithm. Here it goes:
Let’s assume 1<y<x, then
Log_y(x) = k <=> 1 = x/y^k
Log_x(y) =Log_[1/x](1/y) = k’ <=> 1 = 1/y/(1/x)^k’ = x^k’/y = 1
Hence, 1/y^(k-1) = x^(k’-1) <=> 1 = y^(k-1) x^(k’-1)
Taking now logarithms in base y on both sides, and using the rule we prove below on the log of a product, we find
0 = (k-1) Log_y(y) + (k’-1) Log_y(x) = (k-1) + (k’-1) k = -1 + k’ k
1 = k’ k <=> Log_y(x) Log_x(y) = 1 . QED.
Let’s demonstrate the rule for the logarithm of products and quotients.
Be k=n*m, i= iLog_b(n), j=iLog_b(m), all integers. What’s iLog_b(k)? Answer: i+j. Let’s see why. For simplicity sake, we’ll assume that the logaritms are exact. iLog_b(k) corresponds to dividing the rectangle area by b. The side n can be cut by b i times. That still leaves a rectangle of area m. This can be divided by b j times before the resulting area gets <1. Hence the original area can be divided i+j times by b. qed.
Be n, m integers, n divisible by m, i.e., n=m*q, q another integer. What’s iLog_b(n/m)? Answer: i-j. To see why just follow the previous argument changing n.by q and k by n.
There is another way the division and multiplication hinge on 1. In this case the basic idea are paths leading away from 1: The definition of the exponential function b^x for x>0 is 1 multiplied x times by b. If x<0, it means 1 divided x times by b. Clearly, b^0 = 1.
In short, using just the operations of (repeated) multiplication and (repeated) division, given two numbers, b, x, the paths leading towards 1 define the logaritm Log_b(x) while the paths leading away from 1 define the exponentiation b^x.
The idea of paths become more concrete when dealing with complex numbers. There the concept of logarithm is somewhat richer.
It should be interesting to explore this idea of paths for the case of a Group as well as for the case of the other (normed) division algebras. Can it be given a meaningful and useful definition in the case of the Quaternions (Η) and Octonions (Ο)?
The connection to Tarantola’s discussion of the logarithm image of Lie Groups and his geovectors algebra of oriented auto parallel segments as a representation of a group in the neighbourhood of the identity is here somewhat contrived, although it seems to me there is a link.
In any case, I think that, regretably, current high school curricula miss the chance to develop some intuitive understanding of the logarithm. This can be a handicap for studying say the complexity of algorithms or the key concept of information and its relation to logarithms.
More advanced concepts like the Hausdorf dimension can also be easier to grasp with a familiarity of the logarithm as repeated divisions.