In the exact same way as real numbers extend integer points to a continuous line, non-integer powers extend integer powers to a continuous curve. Non-integer powers behave exactly like integer powers, mathematically. So, the best way to intuitively grasp it, is an extension to the integer powers, really.

OK, but there is infinitely many such extensions (even continuous and infinitely differentiable) that differ in the exact curvature between your chosen points. You really need additional arguments to demand this one particular extension that's currently accepted.

No, there is only one the fulfills the standard arithmetic rules of addition and subtraction in the exponent for both integers and non-integers (reals). I mean, for which exp(

*a*+

*b*) = exp(

*a*)·exp(

*b*) and exp(

*a*-

*b*) = exp(

*a*)/exp(

*b*).

So yeah, we really don't have a good analog for non-integer powers. One tripping point is that if we want to calculate the numerical value without having a tool that can do it directly for us, we'll need exponentiation and logarithms to do so. You might think that is because they are somehow special, but it is quite the opposite: exponentiation and logarithms are just the tool we use for the extension, and themselves have a very clear, intuitive definition.

I would agree it's clear and intuitive what 2+2 is, but not sure what's intuitive about the definition of ln(sqrt(2))

I meant that after all the math work to prove their properties are what has been claimed, a layman can grasp exponentiation and logarithm in very simple terms:

exp(

*x*) is the curve that has slope

*x* at

*x*, and has value exp(0)=1.

log

_{e}(

*x*) is the curve that has slope 1/

*x* at

*x*, and has value log

_{e}(1)=0.

You do need lots more math to prove this is so, or to discover why this is so, but for an intuitive understanding of

*what* they are (as opposed to why or how), this suffices.

Unless you define exp in terms of e^{x}, but you can't do that because you just defined a^{b} in terms of exp

Let's not try and be cute here. exp(

*x*) ≝ e

^{x}. I use the form exp(

*x*) to remind the reader that this is not a cyclical definition; that we can define e

^{x} for all real

*x* as a function that does not rely on being able to raise a value to a non-integer power.

It is well and good (and I definitely appreciate it) if you(or anyone else) point out errors or holes in my argument, but in the context of this thread, it is not worthwhile to point out the lack of mathematical rigor, because the issue here is to not prove or claim anything, but to offer A) an intuitive grasp of non-integer powers and B) to show why the need of such intuitive grasp is hindering our use of math as a tool, and why letting go of it, instead understanding that only

*some* of math has a real-world analog or an intuitive explanation and the vast majority is an extension without, is useful and productive.

So, to recap the A) point, in the hopes to clear any confusion, because me fail English often:

We can define non-integer powers like

*a*^{b} = exp(

*b*·log

_{e}(

*a*)). Indeed, computers typically calculate arbitrary real powers

*a*^{b} = 2

^{b·C·log2 a}, where C is a constant (C = log

_{e} 2).

While exp(

*x*) is actually e

^{x}, where e is the number known as Euler's constant (approximately 2.71828), this is not a cyclical argument, because we can define, and understand, exp(

*x*) and log

_{e}(

*x*) differently, as functions that we can define without relying of non-integer powers at all.

Indeed, exp(

*x*) is the curve that has slope

*x* at

*x*, and has value 1 at exp(0).

Similarly, log

_{e}(

*x*) is the curve that has slope 1/

*x* at

*x*, and has value 0 at log

_{e}(1).

(Exactly why and how this is so, requires more math, and goes beyond the scope here.)

For example, if you needed to calculate exp(

*x*) for some

*x* by hand, you could do so by using the series definition, exp(

*x*) = 1 + sum

*x*^{k} /

*k*! for

*k* = 1, 2, 3, ... and so on, where

*k*! is the factorial of

*k* (product of all integers from 1 to

*k*, inclusive).

While we do have some series we can use to calculate log

_{e}(

*x*), they are too slow (need too many terms) to be practical. Instead, we used to use tables of logarithms and exponentials instead,

*y* = exp(

*x*) ⇔

*x* = log

_{e}(

*y*) (listing both

*y* and

*x*), based on pre-calculated exponentials. However, log

_{2}(

*x*) = log

_{e}(

*x*) / C, is much easier to calculate. Indeed, computers typically calculate log

_{e}(

*x*) = C·log

_{2}(

*a*) instead.

(For the exact description of how to calculate log

_{2}(

*x*), see e.g. the Wikipedia

Binary logarithm article. Most computers approximate real numbers with

*floating point* numbers, where

*x* =

*m*·2

^{b}, where

*m* is the mantissa (and normally represents a value between 0.5 and 1.0) and

*b* is the exponent. The binary logarithm is especially powerful for numbers expressed in this form. For example, since 1987 or so, the Intel x86 (and x86-64 or AMD64) processor families have had it in hardware: a machine instruction FYL2X that calculates the binary logarithm of a given value, and another, FYL2XP1, that calculates the binary logarithm of one plus the given value, for even more accurate results near 1.)

To recap point B), most mathematical tools do not really have real-world analogs. We can use integers for counting, and rationals for ratios, but what about irrationals: real numbers that cannot be represented exactly by any ratio? We know π, and it is irrational, so they are definitely useful. Earlier in this thread, I mentioned the Lambert W function, which isn't really a function because we cannot write it in form W(

*x*)=

*something* at all, and belongs to a group called generalized functions, or more properly, distributions (but don't confuse them with statistical distributions; they have some similar properties but are a different thing in math). Many tools in math can be considered

*generalizations* of the things that do have real-world analogs, of the things we can grasp intuitively, but cannot themselves be grasped intuitively (like irrational numbers).

I myself have never been good at math. I might say I'm good at applying the math I know to solve problems, but there are lots and lots of mathematicians that can actually create new mathematical tools! I believe that having the need for real-world analogs was hindering me for years, until I understood and accepted that not everything has such, because they are more like

*extensions*; like using a waldo arm instead of your own arm to manipulate hazardous materials, except for your mind.