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 own value. That is why it owns radioactive decay and capacitor discharge, both of them processes whose rate depends on how much is left. The natural logarithm ln is its inverse, since ln(ex) = x, and that identity is the 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. The one used most often comes out of the second. If N = N0e−λt, then ln N = ln N0 − λt, a straight line in t. Plotting ln N against t is how an exponential gets tested and how its constant gets measured.
Freeing the exponent
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, so e−t/47 = 1.5/6.0 = 0.25.
Take ln of both sides and −t/47 = ln 0.25. Since ln 0.25 = −ln 4, that tidies to t/47 = ln 4.
t = 47 × ln 4 = 65 s. Two halvings, each taking one time constant times ln 2, and the numbers agree: 2 × 47 × 0.693 lands in the same place.
GUIDED PRACTICE
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.
Show the working
Take ln of both sides and ln N = ln N0 − λt.
Plot ln N against t and you get a straight line, gradient −λ, intercept ln N0. A curve on this plot says the decay was never 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.