2.2.2FoundationsStretch

Using Logarithms in Physics

Not every physical relationship gives you a nice straight-line graph. Radioactive decay curves downward, a pendulum’s period grows with the square root of its length, and neither is easy to analyse accurately by eye. Take the logarithm of the right variable, though, and both problems collapse into a straight line — one of the most quietly powerful tricks in the whole course.

35 min Video by A Level Physics HQ Module 2.2 Making measurements and analysing data

Analyse log graphs in physics — A-Level Physics How ToWatch the full walkthrough before the notes below.

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What you'll be able to do

Explain why taking logarithms can turn a curved graph into a straight line
Linearise a power-law relationship $y = A x^{n}$ using logarithms
Linearise an exponential relationship $y = A e^{k x}$ using natural logarithms
Find unknown constants (A, n or k) from the gradient and intercept of a log graph
Identify situations elsewhere in the course where this technique applies

Why logarithms linearise a graph

A straight-line graph is far easier to analyse than a curve — you can read off a gradient and intercept with confidence, and easily judge how well data fits a proposed relationship. Many physical relationships are $not$ naturally linear, but two very common non-linear forms — power laws and exponentials — can both be turned into straight lines by taking an appropriate logarithm of one or both variables.

Tip — The general habit: if a relationship has an unknown POWER, take logs of both x and y. If it has an unknown quantity in an EXPONENT, take the natural log of y only (keeping x as it is).

Linearising a power law

A relationship of the form $y = A x^{n}$ (where $A$ and $n$ are unknown constants) becomes linear when you take logarithms of both sides: $lo g y = lo g A + n lo g x$ . Plotting $lo g y$ against $lo g x$ gives a straight line whose gradient is the power $n$ , and whose $y$ -intercept is $lo g A$ .

lo g y = n lo g x + lo g A

Linearised power law; gradient = n, intercept = log A.

1Gradient

= n = 0.50

, so

T \propto l^{0.5}

— i.e.

T \propto l

, consistent with

T = 2 π l / g

2Intercept

= lo g A = 0.10

, so

A = 1 0^{0.10}

A \approx 1.26

Answern = 0.50 (T ∝ √l); A ≈ 1.26.

Tip — A gradient of exactly 0.5 on a log-log graph is a strong clue you’re looking at a square-root relationship — very common for period-vs-length type formulae in this course.

Linearising an exponential relationship

A relationship of the form $y = A e^{k x}$ becomes linear when you take the $natural$ logarithm of both sides: $ln y = ln A + k x$ . Plotting $ln y$ against $x$ (not $ln x$ — the independent variable here stays as it is) gives a straight line with gradient $k$ and $y$ -intercept $ln A$ .

ln y = k x + ln A

Linearised exponential relationship; gradient = k, intercept = ln A.

1The discharge equation is

I = I_{0} e^{- t / R C}

, so

ln I = ln I_{0} - \frac{t}{R C}

; comparing, gradient

= - \frac{1}{R C}

and intercept

= ln I_{0}

2Intercept:

ln I_{0} = 2.30

, so

I_{0} = e^{2.30} \approx 9.97

3Gradient:

- \frac{1}{R C} = - 0.50

, so

R C = \frac{1}{0.50} = 2.0

Answer

I_{0} \approx 10.0

A; time constant RC = 2.0 s.

Tip — This is exactly the method used later in the course for both radioactive decay (ln N or ln A against t, gradient = −λ) and capacitor discharge (ln Q, ln V or ln I against t, gradient = −1/RC) — learn it once here and reuse it everywhere.

Where this is used elsewhere in the course

This single technique reappears across several later modules: finding a radioactive decay constant from a ln(activity)–time graph, finding a capacitor’s time constant from a ln(charge)–time graph, and testing whether an experimental relationship (for example, between a pendulum’s period and its length, or a resistor’s resistance and temperature) genuinely follows a proposed power law or exponential model, rather than just assuming it does.

Equation recap

lo g y = n lo g x + lo g A

Linearised power law y = Ax^n (gradient = n).

ln y = k x + ln A

Linearised exponential y = Ae^(kx) (gradient = k).

Common mistakes to avoid

Taking the log of both x and y when linearising an exponential relationship.

For y = Ae^(kx), only take the (natural) log of y — x stays as it is on the horizontal axis.

Taking the log of only y when linearising a power law relationship.

For y = Ax^n, BOTH x and y must be logged (log y against log x) — logging only one variable will not give a straight line.

Reading the constant A directly off the graph’s intercept value without "undoing" the logarithm.

The intercept gives log A (or ln A), not A itself — you must raise 10 (or e) to the power of the intercept to recover A.

Key takeaways

Taking logarithms turns awkward non-linear relationships into straight-line graphs that are far easier to analyse.
Power law y = Ax^n: plot log y against log x — gradient = n, intercept = log A.
Exponential y = Ae^(kx): plot ln y against x — gradient = k, intercept = ln A.
This technique underlies later work on radioactive decay constants and capacitor time constants.

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Using Logarithms in Physics

35 min Video by A Level Physics HQ Module 2.2 Making measurements and analysing data