Linearising using Logarithms (DP IB Maths: AI HL)

Revision Note

Author

Dan

Last updated

5 May 2022

Exponential Relationships

How do I use logarithms to linearise exponential relationships?

Graphs of exponential functions appear as straight lines on semi-log graphs
Suppose $y = a b^{x}$
- You can take logarithms of both sides
  - $\ln y = \ln (a b^{x})$
- You can split the right hand side into the sum of two logarithms
  - $\ln y = \ln a + \ln (b^{x})$
- You can bring down the power in the final term
  - $\ln y = \ln a + x \ln b$
$\ln y = \ln a + x \ln b$ is in linear form $Y = m X + c$
- $Y = \ln y$
- $X = x$
- $m = \ln b$
- $c = \ln a$

**How can I use linearised data to find the values of the parameters in an exponential model y = ab^x?**

STEP 1: Linearise the data using $Y = \ln y$ and $X = x$
STEP 2: Find the equation of the regression line of Y on X : $Y = m X + c$
STEP 3: Equate coefficients between $Y = m X + c$ and $\ln y = \ln a + x \ln b$
- $m = \ln b$
- $c = \ln a$
STEP 4: Solve to find a and b
- $a = e^{c}$
- $b = e^{m}$

Worked example

Hatter has noticed that over the past 50 years there seems to be fewer hatmakers in London. He also knows that global temperatures have been rising over the same time period. He decides to see if there could be any correlation, so he collects data on the number of hatmakers and the global mean temperatures from the past 50 years and records the information in the graph below.

2-7-2-a-we-image

Hatter suggests that the equation for $h$ in terms of $t$ can be written in the form $h = a b^{t}$
. He linearises the data using $x = t$ and $y = \ln h$ and calculates the regression line of $y$ on $x$ to be $y = 4.382 - 1.005 x$ .

Find the values of $a$ and $b$ .

2-7-2-ib-ai-hl-exp-relationships-we-solution

Power Relationships

How do I use logarithms to linearise power relationships?

Graphs of power functions appear as straight lines on log-log graphs
Suppose $y = a x^{b}$
- You can take logarithms of both sides
  - $\ln y = \ln (a x^{b})$
- You can split the right hand side into the sum of two logarithms
  - $\ln y = \ln a + \ln (x^{b})$
- You can bring down the power in the final term
  - $\ln y = \ln a + b \ln x$
$\ln y = \ln a + b \ln x$ is in linear form $Y = m X + c$
- $Y = \ln y$
- $X = \ln x$
- $m = b$
- $c = \ln a$

**How can I use linearised data to find the values of the parameters in an power model y = ax^b?**

STEP 1: Linearise the data using $Y = \ln y$ and $X = \ln x$
STEP 2: Find the equation of the regression line of Y on X : $Y = m X + c$
STEP 3: Equate coefficients between $Y = m X + c$ and $\ln y = \ln a + b \ln x$
- $m = b$
- $c = \ln a$
STEP 4: Solve to find a and b
- $a = e^{c}$
- $b = m$

Worked example

The graph below shows the heights, $h$ metres, and the amount of time spent sleeping, $t$ hours, of a group of young giraffes. It is believed the data can be modelled using $t = a h^{b}$
.

2-7-2-b-we-image

The data are coded using the changes of variables $x = \ln h$ and $y = \ln t$ . The regression line of $y$ on $x$ is found to be $y = 0.3 - 1.2 x$ .

Find the values of $a$ and $b$ .

2-7-2-ib-ai-hl-power-relationships-we-solution

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Linearising using Logarithms (DP IB Maths: AI HL)

Revision Note

How do I use logarithms to linearise exponential relationships?

How can I use linearised data to find the values of the parameters in an exponential model y = abx?

How do I use logarithms to linearise power relationships?

How can I use linearised data to find the values of the parameters in an power model y = axb?

You've read 0 of your 5 free revision notes this week

Sign up now. It’s free!

Join the 100,000+ Students that ❤️ Save My Exams

1. Number & Algebra

1.1 Number Toolkit

1.1.1 Standard Form

1.1.2 Approximation & Estimation

1.1.3 GDC: Solving Equations

1.2 Exponentials & Logs

1.2.1 Exponents

1.2.2 Logarithms

1.3 Sequences & Series

1.3.1 Language of Sequences & Series

1.3.2 Arithmetic Sequences & Series

1.3.3 Geometric Sequences & Series

1.3.4 Applications of Sequences & Series

1.4 Financial Applications

1.4.1 Compound Interest & Depreciation

1.4.2 Amortisation & Annuities

1.5 Complex Numbers

1.5.1 Intro to Complex Numbers

1.5.2 Modulus & Argument

1.5.3 Introduction to Argand Diagrams

1.6 Further Complex Numbers

1.6.1 Geometry of Complex Numbers

1.6.2 Forms of Complex Numbers

1.6.3 Applications of Complex Numbers

1.7 Matrices

1.7.1 Introduction to Matrices

1.7.2 Operations with Matrices

1.7.3 Determinants & Inverses

1.7.4 Solving Systems of Linear Equations with Matrices

1.8 Eigenvalues & Eigenvectors

1.8.1 Eigenvalues & Eigenvectors

1.8.2 Applications of Matrices

2. Functions

2.1 Linear Functions & Graphs

2.1.1 Equations of a Straight Line

2.2 Further Functions & Graphs

2.2.1 Functions

2.2.2 Graphing Functions

2.2.3 Properties of Graphs

2.3 Modelling with Functions

2.3.1 Linear Models

2.3.2 Quadratic & Cubic Models

2.3.3 Exponential Models

2.3.4 Direct & Inverse Variation

2.3.5 Sinusoidal Models

2.3.6 Strategy for Modelling Functions

2.4 Functions Toolkit

2.4.1 Composite & Inverse Functions

2.5 Transformations of Graphs

2.5.1 Translations of Graphs

2.5.2 Reflections of Graphs

2.5.3 Stretches of Graphs

2.5.4 Composite Transformations of Graphs

2.6 Further Modelling with Functions

2.6.1 Properties of Further Graphs

2.6.2 Natural Logarithmic Models

2.6.3 Logistic Models

2.6.4 Piecewise Models

3. Geometry & Trigonometry

3.1 Geometry Toolkit

3.1.1 Coordinate Geometry

3.1.2 Radian Measure

3.1.3 Arcs & Sectors

3.2 Geometry of 3D Shapes

3.2.1 3D Coordinate Geometry

3.2.2 Volume & Surface Area

3.3 Trigonometry

3.3.1 Pythagoras & Right-Angled Trigonometry

3.3.2 Non Right-Angled Trigonometry

3.3.3 Applications of Trigonometry & Pythagoras

3.4 Further Trigonometry

**How can I use linearised data to find the values of the parameters in an exponential model y = ab^x?**

**How can I use linearised data to find the values of the parameters in an power model y = ax^b?**