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Exponentials and logarithms

What e and ln actually do, and how to solve for a time buried in an exponent.

The inverse pair

The exponential function ex is its own derivative: its rate of change equals its own value. That is why it describes radioactive decay and capacitor discharge, both processes whose rate is proportional to how much is left. The natural logarithm ln is its inverse, since ln(ex) = x, and that identity is what frees a quantity from an exponent.

Three log rules cover everything A-level asks. ln(ab) = ln a + ln b, ln(a/b) = ln a − ln b, and ln(an) = n ln a. A logarithm is defined only for a dimensionless number, so divide a measured quantity by a fixed reference value of the same unit before taking its logarithm. If N = N0e−λt, then ln(N/N0) = −λt, a straight line in t through the origin with gradient −λ.

A straight log-linear plot is consistent with a single exponential over the measured range; curvature may indicate a different model, changing conditions, background error or measurement limitations.

Freeing the exponent

WORKED EXAMPLE

Solving for a time inside an exponent

A capacitor's pd falls as V = 6.0 e−t/47 (volts, seconds). When does it reach 1.5 V?

Isolate the exponential first, so that the logarithm is taken of a dimensionless ratio: e−t/47 = V/V0 = 1.5/6.0 = 0.25.

Taking ln of both sides gives −t/47 = ln 0.25. Since ln 0.25 = −ln 4, that tidies to t/47 = ln 4.

t = 47 × ln 4 = 65 s. This is two halvings, each taking RC ln 2 with RC = 47 s, which gives the same time because ln 4 = 2 ln 2.

GUIDED PRACTICE

The straight line inside the decay law

Take N = N0e−λt and derive what should be plotted against what to get a straight line, and what its gradient and intercept mean.

Show the working

A logarithm takes a dimensionless argument, so divide by a fixed reference count Nref before taking logs: ln(N/Nref) = ln(N0/Nref) − λt.

Plot ln(N/Nref) against t for a straight line of gradient −λ and intercept ln(N0/Nref). Because N is a count of nuclei, Nref = 1 is usually taken and the axis is labelled ln N. Taking λ from the gradient of the best-fit line uses every point, rather than one pair of readings.

Where physics leans on this: The time constant · Radioactive decay and the decay constant · RP9: capacitor discharge. All eight skills: the maths bridge.