By R. M. Haynes, J. G. Harvey, T. D. Davies (auth.), Robin Haynes (eds.)

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The general form of a logarithmic function is shown in Fig. 20, which corresponds to a relationship in which a unit change in one variable x is associated with a large change in the other variable y at small values, but small changes at large values. Such relationships are fairly common in the environmental sciences. 20 A logarithmic function. EXAMPLE The longitudinal profile of a stream is expressed by: x Hex:: 10gPL where H is the elevation above base level at a distance x downstream of the origin, L is the total stream length, and the base of the logarithm p = L / (1 + L).

This would be of the form y = ax m + bx" + . . c+ Where the highest power of x which appears is the second power, the function is said to be quadratic. For example, y = 3x 2 + 5x + 2 is a quadratic function. 19 Graph of the cubic function y = x 3 - 2X2 - 4x + I. 42 Environmental Science Methods Where x appears to the third power but no higher, the equation is said to be cubic, as in 'y = x 3 - 2X2 - 4x + 1. Plotting the graph of such a function (Fig. 19) and reading off the values of x when y = 0 enables us to solve the cubic equation, x 3 - 2X2 - 4x + 1 = O.

2 Addition of vectors. approximately 730 km towards 330 This is, of course, the same procedure as that involved in constructing a map and then taking measurements and directions from it. In order to indicate that a symbol represents a vector quantity rather than a scalar quantity we will adopt the custom of printing the symbol in bold type. Thus in the example above the displacement of Paris from Milan may be represented by the symbol a and that of Brussels from Paris by b. The displacement of Brussels from Milan may then be written c = a + b.