Note: Some of the content described below is purely speculative and many theories are my personal theories.

From the previous post, it is known that the hyper operators include addition, multiplication, exponentiation, and the hyper 4 operator. However, the other functions such as subtraction are not mentioned. The remaining functions are the inverses of the hyper operators. The inverse of the hyper 1 operator (addition) is subtraction. The inverse of the hyper 2 operator (multiplication) is division. These concepts are simple. However, beyond this, things get more tricky.

The hyper 3 operator is exponentiation. The inverse of exponentiation is known as the logarithm. The logarithm consists of three parts: the base, the argument, and the solution. The standard notation for the logarithm is

logbx=y (pronounced log base b of x equals y)

A log without a base implies base 10.

The equation above implies that b^y=x. Therefore, log28=3 because 2^3=8. Also, logs provide solutions to equations such as 7^x=9 which has no rational solution. Logarithms also have many other properties. The natural logarithm ln(x) is a logarithm with base e (e=2.71828...).

Although I could go on listing the many properties of logs, the point of this post is to introduce a function of my own design, the hyperlogarithm. The hyperlogarithm (written hyperlog(x)) is the inverse of the upper hyper 4 operator. As with logarithms, hyperlogbx=y can only be true if b^^y=x Also similar to logs is that neither a log or a hyperlog can have a negative argument. Just as the hyper 4 operator can be extended to all real numbers, the hyperlog can too (provided that x>0 and b>0 and b is not equal to 1).

Similar to this, one can continue with hyperhyperlogarithms (the inverse of the hyper 5 operator) and so on. This brief post merely outlines the possibilities of the hyperlog and it may have other undiscovered properties.

The equation above implies that b^y=x. Therefore, log28=3 because 2^3=8. Also, logs provide solutions to equations such as 7^x=9 which has no rational solution. Logarithms also have many other properties. The natural logarithm ln(x) is a logarithm with base e (e=2.71828...).

Although I could go on listing the many properties of logs, the point of this post is to introduce a function of my own design, the hyperlogarithm. The hyperlogarithm (written hyperlog(x)) is the inverse of the upper hyper 4 operator. As with logarithms, hyperlogbx=y can only be true if b^^y=x Also similar to logs is that neither a log or a hyperlog can have a negative argument. Just as the hyper 4 operator can be extended to all real numbers, the hyperlog can too (provided that x>0 and b>0 and b is not equal to 1).

Similar to this, one can continue with hyperhyperlogarithms (the inverse of the hyper 5 operator) and so on. This brief post merely outlines the possibilities of the hyperlog and it may have other undiscovered properties.

## 4 comments:

Louis,

I am not able to comment on this post, at least right now, but I may have a query for you to pursue. Something about dating a possible mentioning of an eclipse in Homer's Odyssey. I will give you the article to ponder. Best, Dr. McGay

Just as there can be two inverses to exponentiation, log and root, there can be two inverses to any hyper operator, hyperlog and hyperroot.

This is because functions have inverses, but operators, I mean binary operators with two variables (x^y takes x and y) can be made 'functional' and thus invertible either by holding x constant or y constant.

Your hyper operators really are sort of trinary; ie 'x hyper 4 y' or 'x hyper y z', there are three arguments.

Therefore there should be 3 ways to invert.

Sorry to be replying to this so late, I am reading everything which I discover via google alerts very late because I am much slower at reading the alerts than they are at coming!

Thank you for the comment. But a root, rather than being an inverse of exponentiation, is merely a fractional exponent. For example, the square root of x is the same as x^(1/2) while the cube root of x is the same as x^(1/3). Although the square and the square root cancel each other out, they aren't generally inverse operations. Therefore, the log is the only inverse of exponentiation in its general form.

For the hyper 4 operator, I have attempted to define the hyperlog, although my results are not universally accepted. The hyperroot, however, is simply the hyper 4 operator of its inverse. For example the "hyperroot" of x hyper 4 y is simply x hyper 4 1/y. For each subsequent operator, such as the hyper 5 and hyper 6 operators, there are separate inverses, and I do not believe that the degree of the operator is its own variable.

The last above comment is the first and only place on the internet, I have found, which makes the point that one inverse is enough for a complete formalism of hyperarithmetic. That's what I believe and think, at least. So I'd give you the credit for this.

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