Logarithmic Scales & Transformations — AHL 1.11 Logarithmic scales and transformations in modelling (HL extension) - IB Mathematics Applications and Interpretation

SECTION 1

Historical Context & Motivation

Before calculators existed, multiplying and dividing large numbers was gruelling work. Astronomers and navigators needed a shortcut, and in the early 1600s a Scottish mathematician named John Napier found one: he showed that multiplication could be replaced by addition if you first converted numbers into a new scale. He called his invention logarithms, from the Greek words logos (ratio) and arithmos (number). This single idea shaved years off astronomical calculations and eventually led to the slide rule, the 'calculator' used for over three centuries.

Logarithms didn't stay locked in pure mathematics. Scientists soon discovered that many real-world phenomena — earthquakes, sound intensity, acidity — span such huge ranges that ordinary linear scales are useless. A logarithmic scale compresses these ranges so that patterns become visible. Even more powerfully, when data that follow a power law or exponential model are plotted on a logarithmic axis, the resulting curve becomes a straight line — and straight lines are much easier to analyse.

1614 • Napier publishes logarithms

John Napier releases Mirifici Logarithmorum Canonis Descriptio, introducing tables that convert multiplication into addition.

1624 • Briggs refines common logarithms

Henry Briggs publishes base-10 logarithm tables, making the system practical for everyday computation and navigation.

1935 • Richter scale introduced

Charles Richter develops a logarithmic scale for earthquake magnitude, demonstrating how log scales tame data that varies by factors of millions.

1960s • Log-log plots in science

Researchers across biology, physics, and economics routinely use log-log and semi-log graphs to linearise power and exponential models, making regression analysis far simpler.

Today • IB HL modelling with logarithmic transformations

The IB Applications & Interpretation HL syllabus teaches students to apply logarithmic transformations to real data, connecting centuries-old mathematics to modern data science.

The central question this topic answers is deceptively simple: how can we turn a curve into a straight line? By mastering logarithmic transformations, you gain a toolkit for identifying hidden models in messy data and extracting the parameters that describe them.

SECTION 2

Core Principles & Definitions

Before diving into transformations, you need a solid grip on the building blocks. The ideas below form the foundation of every logarithmic scale and every linearisation technique you'll use in this topic.

Logarithm as Inverse

If b^x = a, then log_b(a) = x. The logarithm answers: 'What exponent turns the base b into a?'

Logarithmic Scale

An axis where equal spacing represents equal multiplicative steps (×10, ×100, …) rather than equal additive steps (+10, +20, …). Each 'tick' is ten times the previous one on a base-10 log scale.

Semi-Log Plot

A graph with one logarithmic axis and one linear axis. Exponential data (y = ab^x) appears as a straight line on a semi-log plot.

Log-Log Plot

A graph where both axes are logarithmic. Power-law data (y = axⁿ) appears as a straight line on a log-log plot.

Linearisation

The process of applying a mathematical transformation (often log) to data so that a non-linear relationship becomes linear. Once linear, you can use y = mx + c to find model parameters.

✦

KEY TAKEAWAY

Think of a logarithmic scale like the volume knob on a stereo. Turning it from 1 to 2 might double the sound energy, and turning from 2 to 3 doubles it again. Each equal step on the dial represents multiplying the quantity, not adding a fixed amount. That's why logarithmic scales are perfect for data that grows by factors rather than by sums.

SECTION 3

Visual Explanation — Linearising an Exponential

The diagram below shows the same data set plotted two ways. On the left is a standard linear-axis graph where exponential growth produces a steep, hard-to-analyse curve. On the right, the y-axis has been replaced with a logarithmic scale (a semi-log plot), and the same data now fall on a straight line. This is the core visual insight of logarithmic transformations.

Left: the exponential function y = 5 × 3^x curves steeply upward. Right: after taking log₁₀ of y, the same data form a straight line with slope log 3 ≈ 0.477 and y-intercept log 5 ≈ 0.699.

Notice how the pink curve on the left rockets upward, making it nearly impossible to read intermediate values or determine the growth rate by eye. On the right, the same five data points fall neatly onto a cyan line. The gradient of that line gives you log b (the logarithm of the base), and the y-intercept gives you log a (the logarithm of the coefficient). From those two numbers, you can recover the original model parameters a and b.

SECTION 4

Mathematical Framework

Linearising a model means applying logarithms so the resulting equation has the form Y = mX + c, where Y and X are transformed variables. The key log laws you need are the product rule log(ab) = log a + log b, the power rule log(aⁿ) = n log a, and the quotient rule log(a/b) = log a − log b.

Linearising an Exponential Model

EXPONENTIAL MODEL

y = a × bˣ

a = initial value (y-intercept when x = 0), b = growth/decay factor, x = independent variable.

TAKING LOG OF BOTH SIDES

log y = log a + x · log b

This is in the form Y = c + mX, where Y = log y, m = log b (the gradient), c = log a (the y-intercept), and X = x. Plot log y against x to get a straight line.

Linearising a Power Model

POWER MODEL

y = a × xⁿ

a = coefficient, n = power/exponent, x = independent variable.

TAKING LOG OF BOTH SIDES

log y = log a + n · log x

This is in the form Y = c + mX, where Y = log y, m = n (the gradient equals the power!), c = log a (the y-intercept), and X = log x. Plot log y against log x to get a straight line.

💡

Which plot tells you which model?

If your data form a straight line on a semi-log plot (log y vs x), the underlying model is exponential. If your data form a straight line on a log-log plot (log y vs log x), the underlying model is a power function. This is the diagnostic trick for choosing the right model.

SECTION 5

Logarithmic Scales in the Real World

Logarithmic scales aren't just a mathematical trick — they are embedded in science and everyday life. Whenever a quantity spans many orders of magnitude (factors of 10), a linear scale becomes impractical. The table below catalogues several famous logarithmic scales, showing how each one compresses a huge range into a manageable number.

Scale	Quantity Measured	Base	Each +1 unit means…
Richter	Earthquake energy	10	Amplitude ×10, energy ×31.6
Decibel (dB)	Sound intensity	10	+10 dB = intensity ×10
pH	Hydrogen-ion concentration	10	−1 pH = H⁺ concentration ×10
Stellar magnitude	Star brightness	≈2.512	+1 magnitude = brightness ÷2.512

Common logarithmic scales and what each unit step represents.

A log-log plot of y = 2x³. Because both axes are logarithmic, the power model becomes a straight line with gradient 3 (the exponent) and y-intercept log 2 ≈ 0.301.

In the log-log plot above, notice that the gradient equals the exponent n = 3 of the original power function. This is extremely useful: if you are given a data set and suspect a power relationship, plot log y against log x, fit a line, and read off the slope to determine the power. The y-intercept gives you log a, so a = 10^intercept.

SECTION 6

Worked Example — Finding a Power Model from Data

A biologist measures the metabolic rate R (in watts) of mammals against their body mass M (in kg). She suspects a power model R = aMⁿ. The data are shown below.

M (kg)	R (W)	log₁₀ M	log₁₀ R
0.1	0.75	−1.00	−0.125
1	3.5	0.00	0.544
10	16	1.00	1.204
100	75	2.00	1.875
1000	350	3.00	2.544

Metabolic rate vs. body mass with log-transformed values.

Determine the power model R = aMⁿ

1
Step 1 — Transform the data

Compute log₁₀ M and log₁₀ R for each data point. These values are already in the table's last two columns.
2
Step 2 — Plot log R against log M

On a log-log plot, the five points should approximate a straight line if the power model is appropriate. Visually, the points do fall near a line, confirming the power model.
3
Step 3 — Find the gradient (slope)

Use two well-separated points: (log M, log R) = (−1.00, −0.125) and (3.00, 2.544). The gradient is m = (2.544 − (−0.125)) / (3.00 − (−1.00)) = 2.669 / 4.00 = 0.667.

n ≈ 0.667 ≈ ²⁄₃
4
Step 4 — Find log a from the y-intercept

Using the point (0.00, 0.544): log R = n × log M + log a → 0.544 = 0.667 × 0 + log a → log a = 0.544.

log a = 0.544, so a = 10^0.544 ≈ 3.50
5
Step 5 — Write the model

Substitute a ≈ 3.50 and n ≈ ²⁄₃ into R = aMⁿ.

R ≈ 3.50 × $M^{2/3}$ — This is remarkably close to Kleiber's law in biology, which states that metabolic rate scales as roughly the ¾ power of body mass.

SECTION 7

Semi-Log vs. Log-Log — When to Use Which

Choosing the right type of logarithmic plot is essential. The table below summarises the differences between semi-log and log-log approaches, linking each to its corresponding algebraic model.

Feature	Semi-Log Plot	Log-Log Plot
Axes	x linear, y logarithmic	Both x and y logarithmic
Model linearised	Exponential: y = a × bˣ	Power: y = a × xⁿ
Linearised form	log y = (log b)x + log a	log y = n · log x + log a
Gradient gives	log b (growth/decay factor)	n (the power/exponent)
y-intercept gives	log a (initial value)	log a (coefficient)
Typical examples	Population growth, radioactive decay, compound interest	Kepler's law, allometric scaling, drag force

Comparison of semi-log and log-log approaches to linearisation.

✦

KEY TAKEAWAY

Imagine you're a detective with two magnifying lenses. The semi-log lens reveals exponential patterns (constant percentage change), while the log-log lens reveals power-law patterns (scaling relationships). Try each lens on your data — whichever one gives a straight line tells you the type of model hiding in the numbers.

SECTION 8

Connection to Advanced Modelling & Regression

In this course, you linearise data and then fit a line by eye or with basic regression on the transformed variables. At a more advanced level, you would use non-linear regression (also called curve-fitting) algorithms that fit the original model directly to the raw data, without needing to transform anything. The table below compares the HL approach with what you might encounter in university statistics or data science.

Aspect	IB HL Approach	Advanced / University Approach
Technique	Linearise with logs, then linear regression	Non-linear least-squares regression on raw data
Tools	GDC, pen-and-paper log tables	Python (scipy), R, MATLAB
Assumptions	Errors are additive in log space	Errors are modelled in original space
Strengths	Visual, intuitive, quick diagnostics	More accurate parameter estimates; handles complex models
Limitation	Log transform distorts error structure	Requires iterative algorithms and larger data sets

IB HL linearisation vs. advanced non-linear regression.

The important takeaway is that the linearisation method you learn now is not a workaround — it is a foundational skill. Understanding why log transformations work gives you the conceptual framework to appreciate more sophisticated regression techniques later. Many professional scientists still use log-log plots as a first step in exploratory data analysis before running computer-based models.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain why plotting log y against x (a semi-log plot) turns the exponential model y = a × bˣ into a straight line. In your explanation, state what the gradient and y-intercept of that line represent in terms of a and b.

Show answer

Taking log₁₀ of both sides of y = a × bˣ gives log y = log a + x · log b. This has the form Y = c + mx, which is the equation of a straight line where Y = log y and X = x. The gradient m equals log b, which tells you the growth or decay factor of the exponential, and the y-intercept c equals log a, which gives the initial value since a = 10^c. The key insight is that the log laws convert multiplication (a × bˣ) into addition (log a + x log b), which is inherently linear in x.

Question 2: BASIC CALCULATION

A data set is linearised on a semi-log plot (log₁₀ y vs x), producing a straight line with gradient 0.301 and y-intercept 1.00. Determine the values of a and b in the exponential model y = a × bˣ.

Show answer

From the linearised form log y = (log b)x + log a, the y-intercept gives log a = 1.00, so a = 10¹ = 10. The gradient gives log b = 0.301, so b = 10⁰·³⁰¹ = 2.00 (since log₁₀ 2 = 0.301). Therefore the model is y = 10 × 2ˣ. You can verify: when x = 0, y = 10 × 1 = 10 (log 10 = 1 ✓), and when x = 1, y = 10 × 2 = 20 (log 20 = 1.301 ✓).

Question 3: INTERMEDIATE

A researcher records the following data and suspects a power model y = axⁿ. x: 2, 5, 10, 20, 50 y: 11.3, 44.7, 126.5, 358, 1414 (a) Compute log₁₀ x and log₁₀ y for each point. (b) Using the first and last transformed points, estimate the gradient n and the coefficient a. (c) Write the power model.

Show answer

(a) log x values: 0.301, 0.699, 1.000, 1.301, 1.699. log y values: 1.053, 1.650, 2.102, 2.554, 3.150. (b) Gradient = (3.150 − 1.053) / (1.699 − 0.301) = 2.097 / 1.398 ≈ 1.50. So n ≈ 1.5 (i.e., y ∝

x^{3/2}

). Using the point (1.000, 2.102): log a = 2.102 − 1.50 × 1.000 = 0.602, so a = 10⁰·⁶⁰² ≈ 4.00. (c) The model is y ≈ 4.0 ×

x^{1.5}

. Check: when x = 10, y = 4 ×

10^{1.5}

= 4 × 31.6 ≈ 126.5 ✓. This confirms the power model with exponent 3/2.

Question 4: APPLIED

The intensity I of light (in lux) passing through glass decreases exponentially with thickness t (in mm): I = I₀ ×

k^t

. A student measures I = 500 lux at t = 0 mm and I = 32 lux at t = 8 mm. (a) By taking log₁₀ of both sides, set up the linearised equation. (b) Find k and I₀. (c) Predict the intensity at t = 5 mm.

Show answer

(a) Taking log: log I = log I₀ + t × log k, which is linear in t. (b) At t = 0: log I₀ = log 500 = 2.699, so I₀ = 500 lux. At t = 8: log 32 = 2.699 + 8 × log k → 1.505 = 2.699 + 8 log k → 8 log k = −1.194 → log k = −0.1493 → k =

10^{−0.1493}

≈ 0.709. So about 70.9% of light passes through each mm. (c) At t = 5: log I = 2.699 + 5 × (−0.1493) = 2.699 − 0.747 = 1.952 → I =

10^{1.952}

≈ 89.5 lux. The log transformation allowed us to find the decay factor without any curve-fitting software.

Question 5: CRITICAL THINKING

A student plots log y vs x and log y vs log x for the same data set. On the semi-log plot, the points curve upward. On the log-log plot, the points form an almost perfect straight line with gradient 2.5 and y-intercept 0.48. (a) Which model type fits this data? Justify your answer. (b) Write the model. (c) The student's friend argues that since R² = 0.97 for a linear fit on the semi-log plot, the exponential model is also acceptable. Critique this argument.

Show answer

(a) The power model y = axⁿ fits this data because the log-log plot gives a straight line, which is the diagnostic test for a power relationship. The semi-log plot curves, ruling out an exponential model. (b) From the log-log line: n = 2.5 and log a = 0.48, so a = 10⁰·⁴⁸ ≈ 3.02. The model is y ≈ 3.0 ×

x^{2.5}

. (c) A high R² on the semi-log plot does not validate the exponential model. Over a limited range, many smooth curves approximate a straight line, inflating R². The correct diagnostic is whether the points are genuinely linear (no systematic curvature), not merely whether R² is high. The curvature on the semi-log plot reveals a systematic misfit that R² alone does not capture. This is why visual inspection of residuals is essential, and why the log-log linearity is the stronger evidence.

SUMMARY

Lesson Summary

Summary

A logarithmic scale replaces equal additive spacing with equal multiplicative spacing, compressing data that span many orders of magnitude. Exponential models (y = a × bˣ) become straight lines on a semi-log plot (log y vs x), while power models (y = a × xⁿ) become straight lines on a log-log plot (log y vs log x). The process of applying logarithms to convert a curve into a line is called linearisation.

Once linearised, the gradient of the line reveals either log b (for exponentials) or n (for power models), and the y-intercept gives log a in both cases. You recover the original parameters by raising 10 to the appropriate power. This technique connects centuries of mathematical history to modern data analysis and forms a key tool in the IB Applications & Interpretation HL toolkit.