The fundamental index laws are given by
If these are true we can also deduce
In this post we discuss the following conditions under which these laws hold.
- Case 1: The laws are true if and are positive real numbers and , are real numbers.
- Case 2: If or is negative we require and to be rational with odd denominator for or to be defined (this includes or being integers as the denominator in this case is 1). All the laws except (2) hold in this case. For (2) to be true we require the denominators of and in reduced form to be odd and that either (a) the numerators and denominators of and are all odd or (b) the numerator of in reduced form is even.
- Case 3: The laws are mostly true if , except need to be positive and laws (6), (7) do not apply.
Case 1: Positive bases
Let us seek to define from first principles for arbitrary positive and real. We can be motivated by the continuity of the function as a function of (where is fixed) and think of as the limit of values as rational numbers approach . To me it is easier to proceed via the logarithm function which may be defined as
Being the area under a continuous positive-valued function, this function is continuous and monotonically increasing. It is also apparent that . From the definition the important identity (where ) can be derived via a change of variable as follows:
By repeated use of this rule, for positive integers . As , grows without bound as increases, which shows that the log function is unbounded above. It is also unbounded below due to the relationship (which follows from ). Hence the log function is monotonic, continuous and has range . It thus has a continuous inverse which is how we may define the exponential function: is the unique positive value satisfying
From the relationship follows
where and are real. Repeated use of this gives
for a positive integer. Defining , we have the familiar form .
We can extend (12) to rational values of . Let us recall what we mean by a rational power. For and relatively prime positive integers we define to be the unique positive number such that . (That the ‘th root exists follows from the fact that the function has an inverse for ).
From (12) we then have
Setting to and to we have thus shown , so . Therefore (12) is satisfied when is positive rational. Finally we use (which can be proved using (11)) to extend (12) to negative rationals:
We are finally ready to define for arbitrary positive and real relying on continuity. We simply extend (12) replacing with (also replacing with ) and with real values :
From this definition it is quick to verify laws (1) and (2):
Law (3) is immediate if either or is equal to 1. Otherwise, we can find so that (choose ) and then we have from (1) and (2) the following:
Case 2: Negative bases
When the base is negative we can define by the real number provided exists. This will be the case if is rational and has odd-valued denominator (square roots of -1 are not real-valued nor are irrational powers of -1). Writing where and are relatively prime ( odd), we then have . Since odd denominators are preserved under addition, law (1) can be shown to hold.
To check law (3) we consider the cases of both being negative or only one (say ) being negative.
a) If and are both negative:
b) If is positive and is negative:
However rule (2) is a little more complicated. For example, if so we cannot say
Issues arise when even denominators in the exponents appear because positive square roots will be taken when negative numbers may be required. It is also possible that but for not to be real-valued (e.g. but is not real-valued).
For to make sense when we require that the left and right sides are defined and are equal in the equality . Hence:
- for the left side to exist, must have odd denominator and if then must also have odd denominator
- for the right side to exist, in reduced form must have odd denominator
- (a) the left side is negative iff both have odd numerators and denominators and the same is true for the right side.
- (b) the left side is positive iff either of has even numerator and the right side is positive iff has even numerator and odd denominator in reduced form.
We conclude that (2) holds when either (a) the numerators and denominators of and are all odd (in which case the result is negative) or (b) the product has even numerator and odd denominator in reduced form (in which case the product is positive).
This allows the possibility of an even numerator cropping up such as . It is interesting to see that is valid but is not for .
Also, reduced form (cancelling out even factors) is important to avoid erroneous calculations such as .
Finally we remark that for negative bases we lose continuity: for example switches between and depending on whether the numerator of is odd or even.
Case 3: Base 0
Note that is for positive (again defined firstly for rationals then extended to positive reals by continuity). However it is undefined for negative which is why the laws only hold for positive indices and neither (5) nor (6) apply. In the discrete world it is common to define to be 1 for convenience but the function fails to be continuous at for any choice of value of .
I’ll end this post with a cool-looking exponential of logarithm identity that is not as well known as it perhaps ought to be. For we have
(The logarithm of both sides is !)
For example, .