The first post of this series, Easy Exponents, laid out the fundamental axiom of exponents and six derived theorems. The second post, More on Negative Exponents, took a deeper look at negative exponents and inverses. This post takes a closer look at non-integer exponents and roots. And the dreaded logarithms.
The fifth theorem from the first post deals with non-integer exponents (specifically, fractions). It says:
For example:
So, fraction exponents of x in the form of one-over-n are the same as taking the nth root of x. And because of the sixth theorem from the first post, we can say:
In general, fraction exponents — rational exponents — can be separated into a root and a regular integer exponent.
Here’s an example done the first way (the second and fourth terms above):
Versus doing it the second way (the third and fifth terms):
Same answer either way. (The “1e6” is just a geeky shorthand for “a one with six zeros after it” — to wit, one million.)
In passing, we can note that:
So, some fraction exponents can be broken into a product of roots. This provides a third solution to the example above:
Which gives us the same answer.
The above is all well and good for rational exponents, but what about irrational exponents? These don’t reduce to any p/q fraction, so how do we interpret those?
The short answer is that these are logarithms — the mathematical opposite of exponents.
The logarithm of some original number, in a given base number, is the exponent that, when applied to the base number, gives us the original number.
For example, given some original number we’ll call N, and assuming a base we’ll call B (I’ll say more about bases below), then:
Where a is the logarithm of N (in base B). In order for exponent a to result in any real value of N (which is must), it must be real itself. Integer and rational exponents can have irrational results. An obvious example is the square root of 2, which is known to be irrational. Of course, this is the same as raising two to the power of one-half — a rational exponent.
It might help to think of it in reverse: There is a set of real numbers whose logarithms are all either rational or integer. A simple example is 100, whose logarithm (in base 10) is 2, an integer. Another example is the real number, 31.6227766… that we kept getting above. Looking at those sequences backwards shows that the logarithm of that real number is 3/4, a rational number.
The base of a logarithm can be any (sane, reasonable) number, but most work with logarithms is done in one of three:
A base of ten because that’s our natural number base. Nearly all the math most people do is in base ten. This is so much the default that when we write
log
— the mathematical symbol for taking the logarithm of a number — and do not specify a base, then base ten is assumed.A base of the special number e because it simplifies a lot of the important math scientists and mathematicians work with. This takes us into more advanced math than I want to cover here. But this use is so important to mathematics that it has its own special symbol,
ln
, rather thanlog
(we call e the “natural log” so think of “ln” standing for “log, natural”).A base of two because Claude Shannon showed us that all information boils down to binary bits — base two. Taking the log-base-two of something gives you the number of bits required to describe that something.
That last sentence suggests something important. The base-two logarithm of a number gives us the number of bits necessary to express the number in binary. Likewise, the base-ten logarithm of a number gives us the number of digits necessary. For example:
Which might give you pause. The first one obviously isn’t right, it takes four digits to express 1,000. And computer wizards know that 256 is 1.0000.0000 in binary — nine bits, not eight. Here’s the deal:
The logarithm of a value is the number of digits (in whatever base applies) required to count up to the value, but not to the value itself.
And it only takes three digits to count to 999 (and only eight bits to count to 255).
There is much more to logarithms, but I’ll leave it here for now.
A final comment. Above with the exponents, and in the previous two posts, x has always been the base number. The fundamental axiom and the derived theorems don’t care about the base. Neither do the theorems of logarithms. What can matter, especially with logarithms, is how humans use them.
So, the base number is entirely a human convenience. We just commonly use base ten because fingers.