Exponential Equations: using e, ln and straight-line graphs
Exponential equations look different in capacitors, radioactive decay and imaging attenuation, but the same idea keeps appearing: a quantity changes by a fixed fraction in each equal step. Natural logs turn that curve into a line.
Work through it in three passes
Understand the shape
Learn why the original physics graph is curved and why taking ln makes a straight line.
Choose the right plot
Match each Module 6 equation to the graph you should draw, including background correction and capacitor charging.
Do the graph task
Calculate logged values, plot them, check the best-fit line, then use the gradient to find the physics constant.
- Recognise equations of the form y = Aekx and y = Ae-kx.
- Use ln to solve for an unknown in an exponential equation.
- Choose what to plot so an exponential relationship becomes a straight line.
- Use the gradient and intercept to find physical constants such as RC, λ and μ.
What an exponential equation is really saying
A Level Physics exponentials usually describe a quantity that changes at a rate proportional to how much of it is left. That is why the graph is curved: the value changes quickly at first, then more slowly as the value gets smaller.
Decay form
Used when a quantity falls towards zero: capacitor discharge, radioactive activity, photon intensity through an absorber.
Approach-to-final form
Used when a quantity rises towards a maximum: capacitor voltage while charging.
In e-t/RC, the quantity t/RC has no unit because RC is a time constant in seconds. In e-λt, λ has unit s-1, so λt has no unit.
The rules you actually use
ln means natural logarithm. It is the inverse of ex, so it is the operation that "undoes" an exponential.
Half left
kx = ln(2)
37% left
After one time constant, a decaying quantity is about 37% of its initial value.
63% reached
After one time constant, a charging capacitor has reached about 63% of its final voltage.
Turning an exponential curve into a straight line
The straight-line equation is y = mx + c. Your job is to make an exponential look like that. For a simple decay equation:
This means if you plot ln(y) on the vertical axis against x on the horizontal axis, the graph should be a straight line with gradient -k and intercept ln(A).
Original graph: curved decay
Linearised graph: straight line
Strictly, logs are taken of dimensionless ratios, such as ln(V / 1 V) or ln(A / 1 Bq). In exam tables you will often see ln(V) for the logged numerical value. The gradient is unchanged, so the physics result is unchanged.
Where this appears in Module 6
Capacitor discharge
Plot ln(V) against t. Gradient = -1/RC, so RC = -1/gradient.
Capacitor charging
Rearrange to E - VC = Ee-t/RC. Plot ln(E - VC) against t.
Radioactive decay
Plot ln(A) against t. Gradient = -λ. Half-life T1/2 = ln(2)/λ.
Count rate with background
Subtract background first. Plot ln(C - Cbg) against t.
X-ray or gamma attenuation
Plot ln(I) against x. Gradient = -μ. Or plot ln(I0/I) against x for gradient +μ.
Medical tracers and PET
Radiopharmaceutical activity decays exponentially. The same graphing method helps find activity, decay constant or effective half-life.
Plot the graph yourself, then check it
Your task
A capacitor is discharged through a resistor. Use the data to test whether the discharge is exponential. Calculate ln(V), plot ln(V) against t, then use the straight-line graph to find RC.
- Fill in the missing ln(V) values to 3 decimal places.
- Press Plot my points to see your transformed graph.
- Press Show expected graph only after you have tried.
- Use the gradient to calculate RC and use the intercept to find V0.
What success looks like
- Your plotted points should lie close to a straight decreasing line.
- The gradient should be about -0.025 s-1.
- The time constant should be about 40 s.
- The intercept should give an initial voltage of about 12 V.
| t / s | V / V | Your ln(V) |
|---|---|---|
| 0 | 12.00 | |
| 20 | 7.28 | |
| 40 | 4.41 | |
| 60 | 2.68 | |
| 80 | 1.62 |
Reveal full answer and marking guide
| t / s | V / V | ln(V) |
|---|---|---|
| 0 | 12.00 | 2.485 |
| 20 | 7.28 | 1.985 |
| 40 | 4.41 | 1.484 |
| 60 | 2.68 | 0.986 |
| 80 | 1.62 | 0.482 |
gradient = (0.482 - 2.485) / (80 - 0) = -0.0250 s-1
For V = V0e-t/RC, gradient = -1/RC, so RC = -1 / -0.0250 = 40 s.
intercept = ln(V0) = 2.485, so V0 = e2.485 = 12.0 V.
Worked examples with graph choices
Example 1: capacitor discharge
A 12.0 V capacitor discharge gives the data below. Show how to find the time constant.
| t / s | V / V | ln(V) |
|---|---|---|
| 0 | 12.0 | 2.485 |
| 20 | 7.28 | 1.985 |
| 40 | 4.41 | 1.484 |
| 60 | 2.68 | 0.986 |
| 80 | 1.62 | 0.482 |
Plot ln(V) against t. The gradient is about:
For discharge, gradient = -1/RC.
The intercept is ln(V0) = 2.485, so V0 = e2.485 = 12.0 V.
Example 2: capacitor charging needs rearranging
A capacitor charges from a 6.0 V supply. You cannot plot ln(VC) against time and expect a straight line.
Start with the charging equation:
Rearrange so the exponential is on its own:
| t / s | VC / V | E - VC / V | ln(E - VC) |
|---|---|---|---|
| 0 | 0.00 | 6.00 | 1.792 |
| 10 | 1.98 | 4.02 | 1.391 |
| 20 | 3.30 | 2.70 | 0.993 |
| 30 | 4.19 | 1.81 | 0.593 |
| 40 | 4.79 | 1.21 | 0.191 |
Example 3: nuclear count rate with background
A radioactive source is measured with a background count rate of 18 counts s-1.
| t / s | Measured C / counts s-1 | Corrected C - Cbg | ln(C - Cbg) |
|---|---|---|---|
| 0 | 238 | 220 | 5.394 |
| 75 | 174 | 156 | 5.049 |
| 150 | 128 | 110 | 4.700 |
| 225 | 96 | 78 | 4.357 |
| 300 | 73 | 55 | 4.007 |
Plot ln(C - Cbg) against t. Do not log the uncorrected measured count rate.
Example 4: attenuation in medical imaging
For X-ray or gamma photons through tissue, intensity often follows I = I0e-μx.
| x / cm | I / % of incident intensity | ln(I / I0) |
|---|---|---|
| 0 | 100 | 0.000 |
| 2 | 69.8 | -0.360 |
| 4 | 48.7 | -0.720 |
| 6 | 34.0 | -1.079 |
Plot ln(I/I0) against x. The line passes through the origin because ln(1) = 0.
This is the same mathematics as radioactive decay: distance through material replaces time, and attenuation coefficient replaces decay constant.
Common mistakes and quick fixes
Logging before rearranging
For charging capacitors, ln(VC) is not linear. Use ln(E - VC).
Forgetting background
For radioactivity, subtract background count rate before taking logs. Otherwise the graph curves and the half-life is wrong.
Losing the minus sign
If the quantity decays, the gradient is negative. The physical constant RC, λ or μ is positive.
Using two rounded points badly
Use a best-fit line over the full plotted range. Two rounded table values can shift the gradient noticeably.
Self-check questions
1. A graph of ln(V) against t for a discharging capacitor has gradient -0.0125 s-1. Find RC.
gradient = -1/RC, so RC = -1 / -0.0125 = 80 s.
2. Which graph should you plot for VC = E(1 - e-t/RC)?
Plot ln(E - VC) against t. The gradient is -1/RC.
3. A corrected count rate halves every 4.0 hours. What is the decay constant?
λ = ln(2) / T1/2 = 0.693 / 4.0 = 0.173 h-1.
4. Intensity follows I = I0e-μx. What does the gradient of ln(I/I0) against x equal?
The gradient is -μ. If you plot ln(I0/I) against x instead, the gradient is +μ.
Final checklist for graphing exponentials
- Identify the physical equation and the independent variable: time, thickness or distance.
- Rearrange so the exponential term is isolated. Subtract background or final value where needed.
- Take natural logs and compare with y = mx + c.
- Label the vertical axis as the logged quantity, for example ln(V) or ln(C - Cbg).
- Use the gradient to find the constant, keeping the sign clear.
- Convert the intercept back with eintercept if you need the initial value.