For y=Axn, with A>0,x>0, logarithm laws give ln y=nln x+ln A. Thus a log-log plot has gradient n and intercept ln A. This works because logarithms turn a variable power into a product. Take logs in one base. Plot log y against log x. Read the gradient as the power; exponentiate the intercept in the same base to obtain the multiplier. If lg y=2lg x+lg3, then y=3x2, not (lg3)x2. At x=2, y=12. Check: logarithmic plots need positive original coordinates. A graph of log y against x, rather than log x, describes an exponential model.

For \(y=Ax^n\), with \(A>0,x>0\), logarithm laws give \(\ln y=n\ln x+\ln A\). Thus a log-log plot has gradient \(n\) and intercept \(\ln A\). This works because logarithms turn a variable power into a product.

Take logs in one base. Plot log \(y\) against log \(x\). Read the gradient as the power; exponentiate the intercept in the same base to obtain the multiplier.

If \(\lg y=2\lg x+\lg3\), then \(y=3x^2\), not \((\lg3)x^2\). At \(x=2\), \(y=12\).

Check: logarithmic plots need positive original coordinates. A graph of log \(y\) against \(x\), rather than log \(x\), describes an exponential model.

Power models on logarithmic axes

Power models on logarithmic axes