Maths bridge › Exponentials and logarithms
Exponentials and logarithms
What e and ln actually do, and how to solve for a time buried in an exponent.
The inverse pair
ex is the function whose rate of change equals its value, which is why it owns radioactive decay and capacitor discharge: processes whose rate depends on how much remains. The natural logarithm ln is its inverse: ln(ex) = x, which is the whole trick for freeing a quantity from an exponent.
Two log rules cover everything A-level asks: ln(ab) = ln a + ln b, and ln(a/b) = ln a − ln b. From the second comes the workhorse: if N = N0e−λt, then ln N = ln N0 − λt, a straight line in t. Plotting ln N against t is how an exponential is tested and how its constant is measured.
Worked and handed over
WORKED EXAMPLE
Solving for a buried time
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: e−t/47 = 1.5/6.0 = 0.25.
Take ln of both sides: −t/47 = ln 0.25. Since ln 0.25 = −ln 4, this is t/47 = ln 4.
t = 47 × ln 4 = 65 s. Two quarterings, each taking one time constant times ln 2, and the numbers agree: 2 × 47 × 0.693 lands in the same place.
YOUR TURN
The straight line hiding inside
Take N = N0e−λt and derive what should be plotted against what to get a straight line, and what its gradient and intercept mean, before opening the working.
Show the working
Take ln of both sides: ln N = ln N0 − λt.
Plot ln N against t: a straight line with gradient −λ and intercept ln N0. A curve on this plot means the decay is not exponential; a line measures λ from every point at once.
Where physics leans on this: The time constant · Radioactive decay and half-life · RP9: capacitor discharge. All eight skills: the maths bridge.