6 Graph-o-log-y or log-graph-y?
I have already hinted how logs can be used to straighten out exponential behaviour. Here we allow an exponential progression of values (y) to be compared on lin-lin or lin-log scales:
Exponential table/graphs
\[v = v_0 r^t\]
| Interation factor (r): | |
| Initial value (v0): | |
| Duration (max t): |
But there is more. Logarithms also allow power-law behaviours to be seen. Here the various possible scales can be seen:
Power law table/graphs
\[v = v_1 t^n\]
| Constant (v1): | |
| Exponent (n): | |
| Range (min,max t): |
The log-log combination gives straight line! The others are not so useful in this case. Also try changing the exponent (even negative values like for the \(1/f^{2}\) behaviour described earlier, corresponding with \(n = -2\)).
We can understand what the log scales are doing here, if we apply logs to the equation \(v = v_1 t^n\):
\[\log v = \log (v_1 t^n) = \log v_1 + n \log t\]
The terms \(\log v_1\) and \(n\) are constant, and \(\log t\) varies. For those still reading, I assume some familiarity with linear graphs, often traditionally cast as \(y = m x + c\) with \(m\) and \(c\) being the “gradient” and “constant” terms, respectively. This is the form of our log-log graph if we cast \(\log v\) onto \(y\), \(\log v_1\) onto \(c\), \(n\) onto \(m\), and finally our \(\log t\) scale onto the variable \(x\) (reversing the order of the sum).