If they can be connected into a line, such as a curve, or zig zag straight line. If they all over the place, you do not.

If you have enough data, you can extrapolate from that, but it is still a presumption, not an exact science.

Plot the log of the function. log of an exponential is a straight line.

Yes and no. Scientifically: You can fit an exponential function to any number of data points, adjusting the exponent and offset as needed. Likewise you can fit a straight line to any data, adjusting the slope and intercept to minimize deviations. With any such function the points might or might not lie closely to the graph. Mathematically: Given a finite number of points, even if they all lie perfectly on the graph of some function, there's still no guarantee that all the points *in between* also do. In other words, tables of values have limitations. You could certainly conclude that some tables can be WELL DESCRIBED by an exponential. Example: (1,2), (2,4), (3,8), (4,16), (5,32).

1) exponential function: "The exponential function is one of many functions in mathematics. The application of this function to a value x is written as exp(x). Equivalently, this can be written in the form e^x, where e is a mathematical constant, the base of the natural logarithm, which equals approximately 2.718281828, and is also known as Euler's number." "As a function of the real variable x, the graph of y=e^x is always positive (above the x axis) and increasing (viewed left-to-right). It never touches the x axis, although it gets arbitrarily close to it (thus, the x axis is a horizontal asymptote to the graph). Its inverse function, the natural logarithm, ln(x), is defined for all positive x. The exponential function is occasionally referred to as the anti-logarithm. However, this terminology seems to have fallen into disuse in recent times. Sometimes, especially in the sciences, the term exponential function is more generally used for functions of the form k*(a^x), where a, called the base, is any positive real number not equal to one. This article will focus initially on the exponential function with base e, Euler's number. In general, the variable x can be any real or complex number, or even an entirely different kind of mathematical object; see the formal definition below." Source and further information: http://en.wikipedia.org/wiki/Exponential_function "But the general principle behind exponential growth is that the larger a number gets, the faster it grows. Any exponentially growing number will eventually grow larger than any other number which grows at only a constant rate for the same amount of time (and will also grow larger than any function which grows only subexponentially)." Source and further information: http://en.wikipedia.org/wiki/Exponential_growth 2) Here is a method to get a perfect fit described: "To fit a total of (n+1) data points we need a polynomial of degree ‘n’." Source and further information: http://arxiv.org/html/math.NA/0203272v1 (See also part II, exact fit in exponential form) 3) "Scientists are often faced with the difficulty of fitting data coming from processes for which they do not have a quantitative prediction. Often, a functional form that fits the data can be found, but it may or may not give insight as to what causes the shape of the data. It's considerably easier to fit data if you start with at least a qualitative prediction of the relationship between your data points. To evaluate another way of fitting the data, try using the math you learned in the population growth problem. " Source and further information: http://mathdl.maa.org/mathDL/4/?pa=content&sa=viewDocument&nodeId=448&bodyId=515 4) Another question is trying to find a non exact fit, where the data points are not exactly on the curve, and some data can be eliminated as outliers. Here some information about this: - Using Matlab for Curve Fitting in Junior Lab: http://web.mit.edu/8.13/matlab/DOCS/fitting-quickstart.pdf - Detecting outliers when fitting data with nonlinear regression: http://www.biomedcentral.com/1471-2105/7/123 - Fitting exponential and power model to data: http://score.kings.k12.ca.us/lessons/snider/Exponential_Model.html - Least Squares Fitting: http://mathworld.wolfram.com/LeastSquaresFitting.html - Fitting data to a straight line, or exponential or power function: http://folk.uio.no/ohammer/past/fitting.html

Xprofessor correctly described the general situation; you can only talk about describing the given table. Quirkie's answer gives you a hint; unfortunately, plotting cannot be considered a precise tool. Here is the math. Some functions have characteristic properties, which allow checking of whether a finite set of pairs of numbers can be generated by the function in question. Linear (more generally, polynomial) and exponential functions are among them. You want to check whether your points follow an exponential relationship y=ka^x. Take logarithms of both parts (the base does not matter, so I'll be using the symbol for the natural logarithms): ln(y)=k+x*ln(a). This is a linear relationship between ln(y) and ln(x), so your problem is reduced to verification of a linear relationship. You do it by means of "divided differences", i.e., expressions of the type (y2-y1)/(x2-x1). Here numbers are merely indexes. If y=ax+b, it is easy to see that all differences would be equal to a constant, namely a, no matter what the values of x and y are. You can go further and calculate differences between the differences, they all will be zeros. The inverse statement is also true: if the differences are constant, then the original relationship cannot be anything but linear. (In general, for a polynomial of the degree N differences of the order N+1 will be zeros; this allows you to find the degree of the polynomial exactly. The definition of the divided difference of the order N is a bit bulky and I don't provide it here, but you can easily find it elsewhere.) Applying this to your original question, after finding logarithms of both numbers in every pair you calculate the divided differences and check whether they are constant. if yes, the logarithms are related linearly and the original function is exponential indeed. In practical applications you have to take into account computational errors. If the deviations from a constant (or from being just zeros for the next order divided differences) cannot be explained by computational errors, you reject the hypothesis of exponentiality. Please note that the described approach is not the same as the least squares approximation by an exponential function, which can be performed always (although not providing an exact fit in most cases).