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September 16

Exponential functions

Take the simple equation of an exponential function.

y = a^x

why can't "a" equal 1 or be a negative real number?

a can be 1 or negative. 1^x is always 1. (-3)^x = (-1)^x3^x, for example. If you're working off the definition of the Exponential function article, we just call exponential functions with base a those with positive base a; it doesn't mean it can't happen. See also on that page "On the complex plane", which goes into more technical reasons for this. Dysprosia 05:35, 16 September 2006 (UTC)

I thought when a = 1 (the equation now being y = 1^x), if this was now drawn onto a graph there would only be a straight line no matter what x equals. Also if the equation becomes y = (-a) ^x, what happens then? Please answer this on a maths B level.(medium level)i don't want to know the technical side.

What's wrong with a straight line? It's a perfectly legitimate function. However, if you have the function (-a)^x = (-1)^xa^x (a positive of course), you will find that it will only be a real number if x is an integer, but that doesn't mean that there's anything really wrong, because we can still find values for the function (that's the advanced bit). Dysprosia 06:25, 16 September 2006 (UTC)

In more detail, consider two specific examples of a.

• If a = 1, then y = a^x becomes y = 1^x, which is the constant function y = 1. This is a function whose graph is a horizontal line, which is both reasonable and useful.
• If a = −1, then for x = 1⁄2 we have y = √(−1). As you may be aware, and can easily verify, no real number squares to −1, because negative times negative yields positive, as does positive times positive. Although we can switch to more sophisticated tools, especially complex numbers, the conclusion is that negative values of a break the definition for most values of x.

This latter issue comes up in the context of one generalization of the Pythagorean distance function, (x²+y²)^1/2. Everywhere a "2" appears, replace it with a real number, p, with the stipulation that p is at least 1. This almost works, but breaks for the reason we just saw. Instead we use

$(|x|^{p}+|y|^{p})^{1/p}.\,\!$

By ensuring non-negativity, we get a well-defined family of interesting distance functions, including Pythagorean distance as the special case p = 2. --KSmrq^T 04:48, 17 September 2006 (UTC)

Actually, if I'm interpreting the question right, if you have a = 1, it's just a straight line and doesn't really exhibit any of the properties of an exponential function. Like others said, if you have negative a, you get weird things happening in-between the "nice numbers". Even if you just look at the integers (using (-2)^x as an example), you get something like "1, (-2), 4, (-8), 16..." which also isn't really so "exponential". —AySz88\^-^ 05:22, 17 September 2006 (UTC)

The only thing that fails for 1^x is the derivative function; everything else holds trivially. You could interpret (-2)^x at the integers to be "exponential-like" in that the terms oscillate exponentially, but that's not really something formally used. Dysprosia 08:36, 17 September 2006 (UTC)

... also the flatness of the graph when a=1 means that the inverse function $x=log_{a}(y)$ is not well-defined for a=1 i.e. you cannot take logs to base 1. Gandalf61 09:55, 17 September 2006 (UTC)

Well, don't forget we're dealing with "medium level" math which is probably pre-calc, and their description of "exponential function" is probably more based on the really representative case. But yeah, it's kinda misleading to completely exclude someting like 1^x from the set of "exponential functions", instead of saying something like "1^x is just a really boring exponential function"... —AySz88\^-^ 19:23, 17 September 2006 (UTC)

a⁰ is equal to one, right? --AstoVidatu 19:52, 17 September 2006 (UTC)

$\lim _{x\to \infty }1^{x}$ and $\lim _{x\to \infty }x^{0}$ are both indeterminate forms and needs to be evaluated. M.manary 21:12, 17 September 2006 (UTC)

Rephrasing the problem to coincide with the domain of definition provided in Exponential function, i.e. that $\!\,a^{x}=e^{x\ln a}$, for a > 0. The requirement that a>0 is equivalent to the requirement that natural logarithm in the definition yields a Real number. This comes to two cases:

Case a = 0... Consider 0^x. As x -> 0, this has the value 0. Contrariwise, a^0 -> 1 as a -> 0. Thus, the function a^x is discontinuous at 0^0 and so the value at 0^0 is a matter of choice. We may associate this behaviour with the essential singularity in the complex logarithm at zero.

Case a < 0... The complex logarithm has a branch cut along the negative real axis. A consequence is that the value of the logarithm may be continuously continued (analytic continuation) from its values along the positive real axis to values along the negative real axis. But, you get different answers if you go clockwise or counterclockwise. So there's not one answer on the negative real axis. Again, there's a choice. Typically, the "resolution" is to switch from the Argand plane (complex plane) to the Riemann surface for the logarithm, which stacks an infinite number of values over each point in the complex plane. This method extends logarithm to a multi-valued function. Then, which value you take for the value of the logarithm on the negative real axis depends on which branch of the Riemann surface you're on -- i.e. whether you continue in the clockwise or counterclockwise direction and how many times you go around zero. (This method of going from one copy of the complex plane to another induces a group structure on the Riemann surface modulo the complex plane. For the logarithm function, this group is infinite and isomorphic to addition on the integers.)

To sum up... We require a > 0 so that we can have the result be single-valued (under the conventional orientation of log's branch point and branch cut). We may extend to all complex a (except zero), but then logarithm generates an infinite number of values per input and the result of the exponentiation is multiple-valued. (... unless we take another step back in generality and don't project the Riemann surface onto the Argand plane.) -- Fuzzyeric 21:20, 17 September 2006 (UTC)

M.manary : No. Both $\lim _{x\to \infty }1^{x}$ and $\lim _{x\to \infty }x^{0}$ are equal to 1. What you meant to say is that $\lim _{x\to a}f(x)^{g(x)}$, where $\lim _{x\to a}f(x)=0$ and $\lim _{x\to a}g(x)=0$ (or $\lim _{x\to a}f(x)=1$ and $\lim _{x\to a}g(x)=\infty$, or $\lim _{x\to a}f(x)=\infty$ and $\lim _{x\to a}g(x)=0$) is indeterminate. Regardless, it is very common to define 0⁰ = 1, so a⁰ = 1 for every a. About a^x for negative a, a natural definition can be given if we restrict ourselves to real x:

$a^{x}=(-a)^{x}(\cos {\pi x}+i\sin {\pi x})\,\!$

-- Meni Rosenfeld (talk) 05:02, 18 September 2006 (UTC)

Nope. I had exactly the limits I wanted. Of course, neither of them were limits at infinity... -- Fuzzyeric 14:41, 18 September 2006 (UTC)

Is there any clearer way to specify that I am addressing User:M.manary than writing "M.manary : " at the beginning? -- Meni Rosenfeld (talk) 15:16, 18 September 2006 (UTC)

<sigh> Sorry. Maybe if it had been linked? -- Fuzzyeric 15:28, 18 September 2006 (UTC)

Fixed :-) -- Meni Rosenfeld (talk) 15:34, 18 September 2006 (UTC)