What is a powerlaw scaling exponent?

The scaling exponent in a powerlaw distribution refers to intuitively the slope of fit on a log10-log10 plot of the data. Why should we care about it? Essentially the scaling exponent tells us how the two variables will change with respect to each other. It is also useful for being able to predict how new samples will fit into an observed distribution or on the other hand be able to spot data points that defy the expected scaling in some interesting way.