Error Propagation In Exponential And Logarithmic Models

Why exponential and logarithmic models need careful error estimates

An input measurement can be uncertain even when a model is known exactly. In an exponential model, a small input change can produce an output change whose size depends on the model’s growth rate. In a logarithmic model, the same input uncertainty can have a larger effect when the input is small.

The derivative measures this local change in scale. It gives a first-order estimate of the output error caused by an input error. The estimate is useful when the input uncertainty is small enough that the model is well approximated by its tangent line over that change.

From input uncertainty to output uncertainty

Let \(y=f(x)\), and suppose the measured input \(x_0\) has a small signed error \(\Delta x\). The actual input is then \(x_0+\Delta x\). Differentiability gives the first-order estimate

The corresponding differential is \(dy=f'(x_0)\,dx\). When \(dx\) represents the input error, \(dy\) estimates the resulting output error. For a worst-case input uncertainty of at most \(\varepsilon\), the first-order estimate of the output uncertainty in absolute value is \(|f'(x_0)|\varepsilon\).

This is an estimate, not an exact identity for a finite input error. The derivative describes the local scale at \(x_0\); if the uncertainty is not small, the linear estimate may be inaccurate.

Exponential models

Consider \(y=Ae^{kx}\), where \(A\) and \(k\) are constants and \(k x\) is dimensionless. Differentiating gives \(y'=kAe^{kx}=ky\). Thus the first-order output error is proportional to the current output:

The second relation is a relative-error estimate: the relative change in the output is approximately \(k\) times the input change. Its sign matters for signed changes; for a worst-case magnitude, use \(|k|\,|dx|\). The parameter \(A\) sets the output scale, while \(k\) determines how strongly an input change affects the output proportionally.

For a model \(y=A b^x\), with \(b>0\), the derivative is \(y'=A b^x\ln b\). Therefore its first-order relative output change is approximately \(\ln(b)\,dx\), provided \(y\ne0\). In particular, a base greater than one gives a positive sensitivity factor, while a base between zero and one gives a negative one.

Logarithmic models

A logarithm requires a positive, dimensionless argument. A common model is \(y=\ln(x/x_{\mathrm{ref}})\), where \(x>0\) and \(x_{\mathrm{ref}}>0\) is a fixed reference value with the same units as \(x\). Since the derivative of \(\ln(x/x_{\mathrm{ref}})\) with respect to \(x\) is \(1/x\), its first-order output error is

This formula estimates absolute uncertainty in the logarithmic output. It does not say that the relative output error is \(dx/x\). If \(y\ne0\), the relative output error is instead estimated by

The denominator shows why relative output error can become large when the nominal logarithmic output is near zero. At a point where the output is exactly zero, relative output error is undefined, although an absolute error estimate can still be useful.

Why the derivative gives the estimate

The justification is the definition of differentiability. For a differentiable function at \(x_0\), the change in its value can be written as

where the remainder satisfies \(r(\Delta x)/\Delta x\to0\) as \(\Delta x\to0\). The derivative term is the part that changes linearly with the input error; the remainder becomes negligible compared with \(\Delta x\) for sufficiently small changes. This is why \(f'(x_0)\Delta x\) is a first-order estimate rather than generally an exact answer.

For \(f(x)=Ae^{kx}\), differentiating gives \(f'(x)=kAe^{kx}=kf(x)\). Dividing the first-order relation by \(f(x_0)\), when \(f(x_0)\ne0\), yields the relative estimate \(dy/y=k\,dx\). For \(f(x)=\ln(x/x_{\mathrm{ref}})\), differentiating gives \(f'(x)=1/x\), so the absolute estimate is \(dy=dx/x\); dividing by the nonzero output gives the relative formula above. Each result follows from the local linear change, with the division step requiring a nonzero nominal output.

A common mistake: confusing input and output relative error

A common error is to assume that a logarithmic model has the same relative-error formula as an exponential model. For \(y=\ln(x/2)\) at \(x=4\), an input change of \(0.04\) has relative magnitude \(0.04/4=0.01\), or \(1\%\). It is incorrect to conclude that the output’s relative error is also \(1\%\).

The correct absolute output estimate is \(|dy|=|dx|/x=0.04/4=0.01\). The nominal output is \(y=\ln(4/2)=\ln 2\approx0.693147\), so its relative output error is

The correction is to differentiate the model first, then divide by the nominal output only if a relative output error is requested and that output is nonzero. For the exponential model \(Ae^{kx}\), relative input and output changes have a particularly simple relationship; for a logarithmic model, absolute output change is tied to relative input change, but relative output change has an additional factor involving the nominal logarithmic value.

Interpreting the estimates responsibly

Error propagation describes how uncertainty is scaled locally; it does not remove uncertainty from the measurement. Before interpreting an estimate, check that the model is being evaluated at the measured input, that the input error is small enough for a first-order estimate to be reasonable, and that all logarithm arguments and exponential exponents are dimensionless as required by the model.

Also distinguish a signed change from an uncertainty magnitude. A signed \(dx\) gives the signed estimate \(dy=f'(x_0)dx\). If only a bound \(|dx|\le\varepsilon\) is known, report the first-order worst-case magnitude \(|f'(x_0)|\varepsilon\), rather than assigning a direction to the output error. Finally, do not report relative output uncertainty when the nominal output is zero; use absolute uncertainty instead.