will have the function: Of course, the inverse is true too: any function of the form will have a straight line as its log–log graph representation, where the slope of the line is m. Finding the area under a straight-line segment of log–log plot To calculate the area under a continuous, straight-line segment of a log–log plot (or estimating an area of an almost-straight line), take the function defined previously and integrate it. Since it is only operating on a definite integral (two defined endpoints), the area A under the plot takes the form Rearranging the original equation and plugging in the fixed point values, it is found that Substituting back into the integral, you find that for A over x0 to x1 Therefore: For m = −1, the integral becomes Applications These graphs are useful when the parameters a and b need to be estimated from numerical data. Specifications such as this are used frequently in economics. One example is the estimation of money demand functions based on inventory theory, in which it can be assumed that money demand at time t is given by where M is the real quantity of money held by the public, R is the rate of return on an alternative, higher yielding asset in excess of that on money, Y is the public's real income, U is an error term assumed to be lognormally distributed, A is a scale parameter to be estimated, and b and c are elasticity parameters to be estimated. Taking logs yields where m = log M, a = log A, r = log R, y = log Y, and u = log U with u being normally distributed. This equation can be estimated using ordinary least squares. Another economic example is the estimation of a firm's Cobb–Douglas production function, which is the right side of the equation in which Q is the quantity of output that can be produced per month, N is the number of hours of labor employed in production per month, K is the number of hours of physical capital utilized per month, U is an error term assumed to be lognormally distributed, and A, , and are parameters to be estimated. Taking logs gives the linear regression equation where q = log Q, a = log A, n = log N, k = log K, and u = log U. Log–log regression can also