Exponents & Logarithms in Modeling — SL 1.6 Exponents and logarithms in modelling (laws; base e and base 10) - IB Mathematics Applications and Interpretation

SECTION 1

Historical Context & Motivation

Long before calculators existed, mathematicians and scientists needed tools to handle numbers that grow or shrink at tremendous rates. Think about how a bacterial colony doubles every hour, or how a radioactive substance loses half its mass every few years. These real-world phenomena pushed brilliant minds to develop exponents and logarithms — two intertwined mathematical ideas that transformed science, navigation, finance, and engineering. Understanding their historical roots helps reveal why these concepts remain indispensable in modern modeling.

1614 • Napier Publishes Logarithms

Scottish mathematician John Napier published his work on logarithms, giving astronomers and navigators a powerful shortcut: they could turn tedious multiplications into simple additions by looking up values in a table.

1624 • Briggs & Common Logarithms

Henry Briggs refined Napier's idea by introducing base-10 (common) logarithms, making calculations with our decimal number system far more practical. His tables were used for centuries.

1683 • Jacob Bernoulli & Compound Interest

While studying compound interest, Jacob Bernoulli discovered that increasing the compounding frequency led to a limiting value — the number we now call e ≈ 2.718.

1748 • Euler Formalises e

Leonhard Euler gave the constant e its name and demonstrated that the natural exponential function eˣ appears everywhere — from population dynamics to radioactive decay — because its rate of change equals itself.

20th C • Modeling the Modern World

Exponential and logarithmic models became standard tools in biology (population growth), physics (radioactive decay), chemistry (pH scale), acoustics (decibels), and geology (the Richter scale). The IB curriculum reflects their universal relevance.

The central question these tools answer is deceptively simple: if a quantity grows (or decays) by a constant percentage in every time period, how do we describe its value at any given moment — and how do we reverse that process to find when a particular value is reached? That 'describing' part uses exponents; the 'reversing' part uses logarithms.

SECTION 2

Core Principles & Definitions

Before diving into modeling, you need a solid grip on the laws that govern exponents and logarithms. These laws are not arbitrary rules to memorize — each one reflects a fundamental property of repeated multiplication and its inverse. Mastering them will let you simplify complex expressions and solve equations that arise in real-world contexts.

Exponent Laws

The key rules: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, and a⁻ⁿ = 1/aⁿ. These let you combine, split, and simplify expressions with the same base.

Logarithm as an Inverse

A logarithm answers: 'What exponent do I need?' If bˣ = y, then log_b y = x. It undoes exponentiation, just as subtraction undoes addition.

Log Laws

Product rule: log(AB) = log A + log B. Quotient rule: log(A/B) = log A − log B. Power rule: log(Aⁿ) = n × log A. These mirror the exponent laws and are essential for solving exponential equations.

Base 10 vs Base e

log₁₀ (written 'log' or 'lg') is used for scales like pH and decibels. logₑ (written 'ln') appears in continuous growth/decay models. Both are available on your GDC and in the IB formula booklet.

Change of Base

To convert between bases: log_a x = ln x / ln a = log x / log a. This formula lets you evaluate any logarithm using the ln or log button on your calculator.

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KEY TAKEAWAY

Think of exponents and logarithms like a volume knob on a speaker. Exponentiation turns the knob — it takes a base and amplifies (or shrinks) it by the power. A logarithm reads the dial — it tells you how many turns were needed to reach a given output. They are two perspectives on the same relationship: bˣ = y ⟺ log_b y = x.

SECTION 3

Visual Explanation — Exponential Growth & Decay

The cyan curve shows exponential growth (y = 20e⁰·³ᵗ): the value starts small and accelerates upward. The pink curve shows exponential decay (y = 200e⁻⁰·³ᵗ): the value starts large and decreases toward zero but never reaches it. Notice that both curves cross the y-axis at their initial amount A₀, and the sign of k determines the direction.

The diagram above captures the essence of exponential modeling. When a quantity changes by a constant percentage per unit time, rather than a constant amount, the result is an exponential curve. Growth curves appear in population studies, viral spread, and compound interest. Decay curves describe radioactive half-lives, cooling objects, and depreciation of assets. In every case, the y-intercept represents the starting value, and the constant k controls how fast the curve climbs or falls.

SECTION 4

Mathematical Framework

The IB SL 1.6 syllabus focuses on two families of models and the algebraic rules that let you manipulate them. Below are the core equations you need. Each one is followed by a note explaining every variable.

EXPONENTIAL MODEL (BASE e)

y = A₀ × eᵏᵗ

y = value at time t; A₀ = initial amount (when t = 0); e ≈ 2.718 (Euler's number); k = continuous growth/decay rate (k > 0 for growth, k < 0 for decay); t = time.

SOLVING FOR TIME USING ln

t = ln(y / A₀) / k

Take the natural log of both sides of y = A₀eᵏᵗ: ln(y/A₀) = kt, then isolate t. This is how you find when a target value is reached.

EXPONENTIAL MODEL (BASE 10)

y = A₀ × 10ᵏᵗ

Used when the model is based on orders of magnitude — common in pH, decibels, and the Richter scale. To solve for t, use log₁₀: t = log(y / A₀) / k.

KEY LOGARITHM LAWS

log(AB) = log A + log B | log(A/B) = log A − log B | log(Aⁿ) = n log A

These three rules work for any base (log₁₀, ln, etc.). They allow you to break apart products, quotients, and powers inside a logarithm, which is essential for isolating unknowns in exponential equations.

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GDC TIP

On your IB-approved GDC, the log button gives log₁₀ and the ln button gives logₑ. If you need another base, use the change-of-base formula: log_a x = ln x ÷ ln a.

SECTION 5

Logarithmic Scales & Their Applications

Logarithms are not just tools for solving equations — they also provide a way to compress huge ranges of data into manageable scales. The Richter scale, the pH scale, and the decibel scale are all base-10 logarithmic scales. Each whole-number step on these scales represents a tenfold (or hundredfold, etc.) change in the actual quantity being measured. This is why a magnitude-7 earthquake is not just 'one more' than a magnitude-6 earthquake — it releases roughly 31.6 times more energy.

On the linear scale (left), the smaller values cluster together and a single large value dominates. On the logarithmic scale (right), the same data is spread more evenly, making trends and patterns easier to see. This is exactly why scientists use log scales for phenomena that span many orders of magnitude.

Scale	Base	What Each Unit Represents	Real-World Use
Richter	10	Each whole number = 10× amplitude increase	Earthquake intensity
pH	10	Each unit = 10× change in H⁺ concentration	Acidity / alkalinity
Decibels	10	Every 10 dB = 10× perceived loudness	Sound intensity
Continuous Growth	e	Used when rate of change is proportional to current value	Population, radioactive decay, finance

Common logarithmic scales and their bases

SECTION 6

Worked Example — Radioactive Decay

A sample of a radioactive isotope has a mass of 80 grams. Its mass decreases according to the model M(t) = 80e⁻⁰·⁰⁵ᵗ, where M is measured in grams and t is time in years. Find (a) the mass after 10 years, and (b) the time it takes for the mass to drop to 20 grams (its quarter-life).

Radioactive Decay — Full Solution

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Step 1 — Identify the model and given values

We are given M(t) = 80e⁻⁰·⁰⁵ᵗ. The initial mass is A₀ = 80 g, and the decay constant is k = −0.05 per year. For part (a), t = 10; for part (b), M = 20.
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Step 2 — Part (a): substitute t = 10

M(10) = 80 × e⁻⁰·⁰⁵ ˣ ¹⁰ = 80 × e⁻⁰·⁵. Using a calculator, e⁻⁰·⁵ ≈ 0.6065. Therefore M(10) = 80 × 0.6065.

M(10) ≈ 48.5 grams
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Step 3 — Part (b): set M = 20 and solve for t

20 = 80e⁻⁰·⁰⁵ᵗ. Divide both sides by 80: 20/80 = e⁻⁰·⁰⁵ᵗ, so 0.25 = e⁻⁰·⁰⁵ᵗ.
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Step 4 — Take the natural logarithm of both sides

ln(0.25) = −0.05t. We know ln(0.25) = ln(1/4) = −ln 4 ≈ −1.3863. So −1.3863 = −0.05t.
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Step 5 — Isolate t

Divide both sides by −0.05: t = −1.3863 / (−0.05) = 27.726.

t ≈ 27.7 years — the sample reaches one-quarter of its original mass after about 27.7 years.

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CHECK YOUR ANSWER

Substitute t = 27.7 back: M = 80e⁻⁰·⁰⁵ ˣ ²⁷·⁷ = 80e⁻¹·³⁸⁵ ≈ 80 × 0.2502 ≈ 20.0 ✓. Always verify by plugging your answer back into the original equation.

SECTION 7

Strengths & Limitations of Exponential Models

Exponential models are powerful, but no model is perfect. Understanding when these models work well and when they break down is a key part of mathematical literacy — and a topic the IB frequently tests.

Strengths	Limitations
Captures constant-percentage change accurately (e.g., 5% per year compounding).	Assumes the growth/decay rate k stays constant forever — unrealistic for long time frames.
Only two parameters (A₀ and k) make it easy to fit to data with regression.	Exponential growth predicts infinite values, but real populations hit resource limits (logistic model is more realistic).
Works across many disciplines: biology, physics, finance, chemistry.	The model never reaches zero in decay — it predicts an infinitely small but non-zero remainder.
Log transformation linearises the data, making it easy to verify fit.	Sensitive to the initial conditions — a small error in A₀ or k can compound over time.

When to trust — and when to question — exponential models

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KEY TAKEAWAY

An exponential model is like a weather forecast: it is excellent in the short term but becomes less reliable the further out you project. In an IB context, always state the domain of validity — the range of t values for which the model is reasonable — and note any simplifying assumptions.

SECTION 8

Connection to Advanced Models

The exponential model you learn at SL is a stepping stone to more sophisticated tools you may encounter at HL or in university. Seeing how these ideas extend helps you appreciate why the SL foundation matters.

Feature	SL 1.6 Exponential Model	HL / University Extension
Growth pattern	Unlimited exponential growth y = A₀eᵏᵗ	Logistic growth y = L / (1 + Ce⁻ᵏᵗ) with carrying capacity L
Rate of change	Constant relative rate k	Differential equations: dy/dt = ky(1 − y/L)
Data fitting	Log transformation + linear regression	Nonlinear regression, least squares optimisation
Multiple variables	Single independent variable t	Systems of differential equations with multiple coupled variables

SL exponential model vs. advanced extensions

Even at SL, recognising that exponential models have boundaries prepares you for the more nuanced thinking expected at higher levels. When an IB question asks you to comment on the validity of a model, you can reference limits such as carrying capacity, resource constraints, or the assumption that the rate k remains constant. This kind of critical commentary often earns top marks on Paper 3 investigations.

SECTION 9

Practice Problems

Question 1: CONCEPTUAL

Explain, in your own words, why the equation log_b y = x is equivalent to bˣ = y. Why do we call the logarithm the 'inverse' of an exponent?

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The statement bˣ = y tells us that the base b raised to the power x gives y. The logarithm asks the reverse question: 'What power must I raise b to in order to get y?' The answer is x, so log_b y = x. We call it the inverse because it 'undoes' exponentiation, just as division undoes multiplication. If you apply the exponential and then the log (or vice versa), you return to your starting value.

Question 2: BASIC CALCULATION

Simplify the expression: log₁₀(1000) + log₁₀(0.01) − log₁₀(10). Give your answer as an exact integer.

Show answer

Using the definition of log₁₀: log₁₀(1000) = 3 because 10³ = 1000. log₁₀(0.01) = −2 because 10⁻² = 0.01. log₁₀(10) = 1 because 10¹ = 10. Therefore the expression equals 3 + (−2) − 1 = 0. This also makes sense by the product/quotient rules: log(1000 × 0.01 / 10) = log(10/10) = log(1) = 0.

Question 3: INTERMEDIATE

A population of bacteria is modeled by P(t) = 500e⁰·⁰⁸ᵗ, where t is in hours. (a) Find the population after 12 hours, to the nearest whole number. (b) Find how long it takes for the population to reach 2000.

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(a) P(12) = 500e⁰·⁰⁸ˣ¹² = 500e⁰·⁹⁶ ≈ 500 × 2.6117 ≈ 1306 bacteria. (b) Set 2000 = 500e⁰·⁰⁸ᵗ → 4 = e⁰·⁰⁸ᵗ → ln 4 = 0.08t → t = ln 4 / 0.08 ≈ 1.3863 / 0.08 ≈ 17.3 hours. Always verify: 500e⁰·⁰⁸ˣ¹⁷·³ ≈ 500 × 3.997 ≈ 1999, confirming the answer.

Question 4: APPLIED

The pH of a solution is defined as pH = −log₁₀[H⁺], where [H⁺] is the hydrogen ion concentration in moles per litre. (a) If orange juice has [H⁺] = 3.2 × 10⁻⁴ mol/L, find its pH. (b) If bleach has a pH of 12.5, find [H⁺].

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(a) pH = −log₁₀(3.2 × 10⁻⁴) = −[log₁₀(3.2) + log₁₀(10⁻⁴)] = −[0.505 + (−4)] = −(−3.495) ≈ 3.49. (b) 12.5 = −log₁₀[H⁺] → log₁₀[H⁺] = −12.5 → [H⁺] = 10⁻¹²·⁵ ≈ 3.16 × 10⁻¹³ mol/L. Notice how a change of just a few pH units corresponds to a change of many orders of magnitude in concentration — this is the power of the logarithmic scale.

Question 5: CRITICAL THINKING

A student models the number of active users on a new app with N(t) = 200e⁰·¹⁵ᵗ, where t is in weeks. After 20 weeks, the app has only 2400 users instead of the model's prediction. (a) What does the model predict at t = 20? (b) Suggest at least two reasons why the model over-predicts. (c) Propose a more realistic type of model and explain your reasoning.

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(a) N(20) = 200e⁰·¹⁵ˣ²⁰ = 200e³ ≈ 200 × 20.09 ≈ 4018 users, which is significantly higher than the observed 2400. (b) Possible reasons: (i) the growth rate k = 0.15 may have decreased over time as the initial novelty wore off, violating the constant-rate assumption; (ii) the potential user base is finite (market saturation), so unlimited exponential growth is unrealistic; (iii) competitors or negative reviews may have slowed adoption. (c) A logistic model N(t) = L / (1 + Ce⁻ᵏᵗ) would be more appropriate because it includes a carrying capacity L representing the maximum possible users, causing growth to slow as N approaches L. This reflects real-world market saturation.

SUMMARY

Lesson Summary

Summary

In this lesson you learned that exponents describe quantities that grow or decay by a constant percentage per time period, while logarithms are their inverse — they 'undo' exponentiation to solve for unknown exponents. The core model y = A₀eᵏᵗ uses base e ≈ 2.718 for continuous processes, while base-10 models appear in logarithmic scales such as pH, decibels, and the Richter scale. The exponent laws (product, quotient, power) and the corresponding logarithm laws let you simplify expressions and isolate unknowns in equations.

To solve for time in a growth or decay problem, take the natural logarithm (ln) of both sides and apply the formula t = ln(y / A₀) / k. Always state assumptions and limitations — exponential models assume a constant rate k and can over-predict in the long run. The change-of-base formula (log_a x = ln x / ln a) lets you convert between any bases using your GDC. Mastering these tools prepares you not only for IB exams but also for modelling real-world phenomena in science, economics, and beyond.