Exponentials & Logarithms

Previous section: Integration

Next Section: AS Pure General Support

Word Doc

PDF

AS Maths – Pure – Intro to Exponential Functions

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

14A Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video (Up to 2:21)

Bicen Maths Live Lesson

Exponentials & Logarithms 1 • What are exponential functions? • P1 Ex14A • 🤖

Exponentials & Logarithms 2 • Sketching/Transforming Exponential Graphs • P1 Ex14A • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 1-7✦

Extra Questions

• Dr Frost Questions (Very Short, Mainly Transformations)

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 423b: Complete a table of values for an exponential graph with equation y=ka^x for positive integers a and k
• 423e: Plot an exponential graph of the form y = ab^x or y = ab^-x using a completed table of values.
• 423g: Complete a table of values and plot the corresponding exponential graph.
• 424a: Identify the y-intercept of an exponential graph given in the form y=a^x
• 424b: Identify the y-intercept of an exponential graph given in the form y=ka^x
• 424e: Identify the y-intercept of an exponential graph given in the form y=ka^x +b or y=ka^-x + b
• 424f: Identify asymptotes given an exponential graph.
• 424g: Identify asymptotes given the equation of an exponential graph of the form y=ka^x + b for positive integer a
• 424h: Identify asymptotes given the equation of an exponential graph of the form y=ka^-x + b
• 424i: Identify asymptotes given the equation of an exponential graph of the form y=ka^x + b for fractional a
• 423: Exam Practice: Plotting exponential graphs
• 424: Exam Practice: Properties of exponential graphs

Exponential Graphs

Corbettmaths

Exponential growth and epidemics

3Blue1Brown Covid 19 Exponential Graph created March 2020

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• thechalkface Exponential Growth Graph
• thechalkface Logarithms Intro Good Starter Activity from Exponentials to Needing Logs. Could be Introduced at 14D
• Math = Love (Sarah Carter)
  • Growth & Decay Foldable
  • No Fives Dice Decay Activity
  • Skittles Growth & Decay Activity
• Illustrative Mathematics
  • Decaying Dice
• Desmos
  • Marbleslides: Exponentials (transform exponential graphs so the marbles pass through the stars)
• Underground Mathematics
  • Logs and Exponentials: Exponential graphs✦
• Ticktock Maths
  • Exponential Functions lesson (slides)

Worksheets

• kutasoftware Drawing Exponentials
• MadAsMaths (T. Madas)
  • Logarithms Exam Questions Q10, 44, 46, 62, 79, 86 (difficulty 3–4 of 5)

AS Maths – Pure – y=e^x

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

14B Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video (2:21 to 11:40) (Sketching only)

Bicen Maths Live Lesson

Exponentials & Logarithms 3 • Euler’s Number, Differentiating e^x • P1 Ex14B • 🤖

Exponentials & Logarithms 4 • More Graph Sketching • P1 Ex14B • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 8-13✦

Extra resources

• JFYI – What is e and why is it so important?

Extra Questions

• Dr Frost Questions (Short)

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 528a: Recognise Euler’s number e and its equivalent definitions, including Bernoulli’s compound interest problem.
• 526a: Sketch graphs given in the form y = a e^bx for positive b
• 526b: Sketch graphs given in the form y = a e^bx for negative b
• 526c: Sketch graphs given in the form y = a e^{ax + b} + c
• 528: Exam Practice: Solving equations and manipulate expressions involving e^x and ln(x)
• 526: Exam Practice: Graphs of exponential functions y = e^ax+b + c and exponential growth and decay

Other Videos & Multimedia

e (Euler's Number) – Numberphile

Courtesy of Numberphile (see extension activity in 14F after introducing logs)

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• The Chalkface
  • Compound Interest: PDF
• Desmos
  • Exploring e (compound interest at increasing frequency leading to e)
• Ticktock Maths
  • Y=e^{kx} lesson (slides)

AS Maths – Pure – Modelling with Exponential Functions

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

14C Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video (From 11:40) (Sketching only)

Bicen Maths Live Lesson

Exponentials & Logarithms 5 • Simple Exponential Modelling • P1 Ex14C • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 14-18✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 424c: Interpret the value of k in an exponential model y = ka^x in context.
• 424d: Interpret the value of a in an exponential model y = ka^x in context.
• 526d: Determine the output of an exponential function in context.
• 526e: Determine the initial value of an exponential model involving time.
• 526f: Determine the long-term limit of an exponential model involving time.
• 528l: Use a modelled scenario to determine the parameters in an exponential model y = a + be^kx
• 424: Exam Practice: Properties of exponential graphs
• 526: Exam Practice: Graphs of exponential functions y = e^ax+b + c and exponential growth and decay
• 528: Exam Practice: Solving equations and manipulate expressions involving e^x and ln(x)

Further Practice

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Math = Love (Sarah Carter)
  • Cooling Corpse Task
• The Chalkface
  • Carbon Dating: PDF
  • Over The Moon: PDF
• Ticktock Maths
  • Exponential modelling lesson (slides)

AS Maths – Pure – Logarithms

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

14D Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video: Up to 7:07 · For Sketching logs up to 4:56

Bicen Maths Live Lesson

Exponentials & Logarithms 6 • What are logarithms? • P1 Ex14D • 🤖

Exponentials & Logarithms 7 • Graphs of Logarithmic Functions • P1 Ex14D • 🤖

Exponentials & Logarithms 8 • Using your calculator • P1 Ex14D • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 19-24✦

Extra resources

• Sherlocked C2 Logarithms 1

Exam Practice

• Exam Practice (Mainly Sketching Graphs, Q2 and 7 use ln)

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 527a: Evaluate the log of a value (without a calculator).
• 527b: Convert from index form to logarithmic form.
• 527c: Solve logarithmic equations given in the form log_ax=b
• 527: Exam Practice: Laws of logs (excluding ln(x))

Other Videos

Logarithms

Courtesy of Maths Genie

Enrichment Video

Log Tables – Numberphile

Numberphile: Log Tables

Triangle of Power

3Blue1Brown: Triangle of Power

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Online Platforms (with Videos & Exercises)

• Khan Academy
  • Lesson

Activities

• Berwick Maths Diagnostic Questions
• WW Logs Animation
• thechalkface Logarithms Intro Good starter activity from exponentials to needing logs.
• Math = Love (Sarah Carter)
  • Evaluating Logarithms Auction
  • Exponential & Log Form Foldable
  • Logarithms Foldable
  • Logarithm Bingo
  • Log War Activity
  • Logarithm Speed Dating
  • Introducing Logarithms Activities
• Lisa Henry
  • Log War Cards
• Transum
  • Logarithms

Worksheets

• kutasoftware
  • Exponentials & Logarithms 1 · 2
  • Graphing Logs

AS Maths – Pure – Laws of Logs

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept. Video Covers 13E only (Area under the curve)

Download Blank Notes

Download Slides

14E Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video (From 7:07)

Bicen Maths Live Lesson

Exponentials & Logarithms 9 • Laws of Logs – proof • P1 Ex14E • 🤖

Exponentials & Logarithms 10 • Laws of Logs – examples • P1 Ex14E • 🤖

Exponentials & Logarithms 11 • Simplifying ln expressions • P1 Ex14E • 🤖

Exponentials & Logarithms 12 • Solving Equations with Logarithms • P1 Ex14E • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 25-31✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 527f: Use laws of logs to simplify a logarithmic expression.
• 527g: Use laws of logs to write a logarithm as an expression by substitution.
• 527h: Solve logarithmic equations given in the form log[textf(x)]=log[textg(x)]
• 527i: Solve equations using logarithm product and quotient laws (excluding power law).
• 527j: Solve logarithmic equations by using the power law.
• 527k: Use laws of logs to make an algebraic variable the subject.
• 527o: Use the change of base formula for logs.
• 527: Exam Practice: Laws of logs (excluding ln(x))

Other Videos

The Rules of Logarithms

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Online Platforms (with Videos & Exercises)

• Khan Academy
  • Properties of Logarithms

Activities

• Berwick Maths Diagnostic Questions
• thechalkface Summary Page for Logs and Their Rules
• Standards Unit Simplifying Logarithmic Expressions✦
• Andy Lutwyche
  • Codebreaker
• Math = Love (Sarah Carter)
  • Swap Two task
  • Logarithm Tarsia Puzzle
  • Change of Base Notes
  • Properties of Logarithms Foldable
• variationtheory.com
  • Logs and Exponentials: Logarithms✦

Worksheets

• Jethwa Maths Starter Activities (p1-4 & 11-12)
• kutasoftware
  • Logs Inverses (requires power law) Could be good way to link from last section)
  • Properties of Logs
• Mathshelper (Jon Stone)
  • Logarithms Theory
  • Logarithms
• Physics & Maths Tutor
  • Laws of Logarithms: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• SRWhitehouse
  • Logs worksheets✦
• Naiker Maths
  • Logs exam questions (solutions)✦
• Dr Oliver Mathematics
  • Logs questions✦ (solutions✦)
• MadAsMaths (T. Madas)
  • Logarithms Exam Questions Q1–5, 11, 13, 19, 25, 33, 41–43, 45, 51, 53, 55–58, 105, 116, 119 (difficulty 2–5 of 5)
  • Logarithms Practice Simplifying Expressions Q1–7
  • Exponentials & Logarithms Exam Questions Q10, 19, 33–34, 62, 76, 112 (difficulty 2–5 of 5)
  • Exponentials & Logarithms Practice Q7
• Starting Points Maths
  • Basic Logarithm Practice and Guided Discovery of Key Rules

AS Maths – Pure – Solving Equations using Logarithms

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

14F Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video (from 12:21)

Bicen Maths Live Lesson

Exponentials & Logarithms 13 • Solving Equations with Exponentials • P1 Ex14F • 🤖

Exponentials & Logarithms 14 • “Quadratic” Exponential Equations • P1 Ex14F • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 32-36✦

Extra Resources

• Sherlocked C2 Logarithms 2

Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 527d: Solve equations given in the form a^x = b using logarithms.
• 527e: Determine the x-intercept of an exponential graph.
• 527l: Solve equations which are quadratic in an exponential term (excluding e^x).
• 527m: Solve simultaneous equations involving logarithms.
• 527n: Solve equations given in the form a^f(x) = b^g(x), for linear exponents, using logarithms.
• 358d: Solve equations by transforming to the same base, leading to a linear equation.
• 527: Exam Practice: Laws of logs (excluding ln(x))
• 358: Exam Practice: Expressing a power using a different base

Further Practice

• Practise Questions

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Activities

• Berwick Maths Diagnostic Questions
• Math = Love (Sarah Carter)
  • Solving Log Equations Foldable
  • Solving Exponential Equations Foldable
  • Exponential Equations with Logs

Worksheets

• Jethwa Maths Starter Activities (p9-10)
• thechalkface Booklet Including a Very Nice ‘Largest Prime’ Q at the Bottom of Page 3
• kutasoftware Logarithm Equations, Simple
• Physics & Maths Tutor
  • Exponential Equations: Questions 2021–24 · Mark scheme · Model answers · Questions 2018–21 (with MS) · Model answers
• MadAsMaths (T. Madas)
  • Logarithms Exam Questions Q6–9, 12, 14–18, 20–21, 23–24, 26–32, 34–37, 39–40, 47, 49–50, 52, 54, 59–61, 63, 65–67, 69–78, 81–85, 87–91, 93–95, 97–103, 110–112, 115, 118 (difficulty 2–5 of 5)
  • Logarithms Practice Solving Equations Using Logarithms Q1–8; Solving Equations Involving Logarithms Q1–3
  • Equations: Advanced Skills Q13, 20, 24, 28 (difficulty 3–4 of 5)
  • Exponentials & Logarithms Exam Questions Q13, 52, 84, 92 (difficulty 3–4 of 5)

Extension/ Further Interest

• kutasoftware Worksheet (after students watch numberphile video in 14B) Compound Interest Problems. Give answer to first Q.

AS Maths – Pure – Natural Logarithms

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

14G Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video (Graphing ln from 4:56)
• Topic Video

Bicen Maths Live Lesson

Exponentials & Logarithms 15 • More Natural Logarithms (ln) • P1 Ex14G • 🤖

Exponentials & Logarithms 16 • Modelling pt. 1 • P1 Ex14 ExamQs • 🤖

Exponentials & Logarithms 17 • Modelling pt. 2 • P1 Ex14 ExamQs • 🤖

Exponentials & Logarithms 18 • Coming up with your own model • P1 Ex14 ExamQs • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 37-40✦

Exam Practice

• Exam Practice

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 528b: Solve exponential equations involving a single occurrence of e
• 528c: Solve logarithmic equations involving a single occurrence of ln
• 528d: Determine the value of a physical quantity, e.g. time, in an exponential model.
• 528e: Determine the amount of time for a population to reach a multiple of the initial population.
• 528f: Determine the value of a constant in an exponential model using known quantities.
• 528g: Use laws of logs to solve equations involving ln
• 528h: Solve equations which are quadratic in e^x
• 528i: Solve equations which are quadratic in e^x after manipulation.
• 528j: Determine the point of intersection of two exponential equations with different powers.
• 528k: Solve equations given in the form a^x e^x = k
• 529a: Sketch y = log(x) or y = ln(x) (not yet active)
• 529b: Determine coordinates of intercepts with the axes for functions given in the form y = log(x + a) or y = ln(x+a) (not yet active)
• 529c: Sketch graphs given in the form y = log(x + a) or y = ln(x + a) (not yet active)
• 529d: Sketch graphs given in the form y = log(a – x) or y = ln(a – x) (not yet active)
• 529e: Sketch graphs given in the form y = log(ax+b) or y = ln(ax+b) (not yet active)
• 528: Exam Practice: Solving equations and manipulate expressions involving e^x and ln(x)
• 529: Exam Practice: Graphs of y = log_a(x) and y = ln(x)

Further Practice

• Practise Questions

Other Videos

The Exponential Function e and The Natural Log ln

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Online Platforms (with Videos & Exercises)

• Khan Academy
  • Introduction to e and ln
  • Solving Exponential Equations with Logarithms

Activities

• Berwick Maths Diagnostic Questions
• Ticktock Maths
  • Natural Logarithms lesson (slides)

Worksheets

• Jethwa Maths Starter Activities (p5-8)
• Solomon C3 Worksheet✦ and Solutions✦
• kutasoftware
  • Equations with Logarithms
  • Logarithmic Equations, Hard
• MathedUp
  • Natural logarithms: Challenge · Answers
• Naiker Maths
  • Exponentials & Natural Logarithms: Exam Qs 2005–11✦ · Mark scheme✦ · Exam Qs 2012–17 · Mark scheme · Extra practice · Solutions
• MadAsMaths (T. Madas)
  • Exponentials & Logarithms Exam Questions Q1, 3, 6, 8, 12, 14, 20, 22, 24, 26–32, 36, 38, 40, 42, 45–48, 53–54, 56–58, 61, 63–68, 73, 77–79, 83, 87, 89–90, 94, 96, 98, 105, 108, 113–114, 116, 120, 123–125 (difficulty 2–5 of 5)
  • Exponentials & Logarithms Practice Q1–6, 8–15

AS Maths – Pure – Using Logarithms in Non-Linear Data

Courtesy of Owen134866 and Haberdashers’ Adams Maths Dept.

Download Blank Notes

Download Slides

14H Slides (Courtesy of Owen134866)

Dr Frost Video

• Topic Video

Bicen Maths Live Lesson

Exponentials & Logarithms 19 • Logs and Non-Linear Data – real life applications • P1 Ex14H • 🤖

Exponentials & Logarithms 20 • Logs and Non-Linear Data – the theory • P1 Ex14H • 🤖

Exponentials & Logarithms 21 • Logs and Non-Linear Data – examples • P1 Ex14H • 🤖

Live Lesson Using Dr Frost Based Slides Courtesy of Bicen Maths

Download Blank Notes

(Courtesy of King’s Chester Maths Department)

Download Slides

• Dr Frost Slides 41-48✦

Full Coverage

• Full Coverage: Modelling with Exponential Equations (Word)
• Full Coverage: Modelling with Exponential Equations (PDF)

Extra Questions

• Dr Frost Questions

Key Skills

Note all numbers are subject to change and will be updated once all key skills have been finished by Dr Frost

• 530a: Determine the parameters of an exponential graph y = ab^x using the equation of the line of best fit of the logged data.
• 530b: Determine the parameters of polynomial graph y = ax^b using the equation of the line of best fit of the logged data.
• 530c: Use logarithms to estimate parameters of graphs with equations in the form y=ab^x given the data points.
• 530d: Use logarithms to estimate parameters of graphs with equations in the form y=ax^b given the data points.
• 530: Exam Practice: Using log graphs to estimate parameters in y = ax^b and y = ab^x for given data in x and y

Further Practice

• Practise Questions

Other Videos

Exponential Growth and Decay

Courtesy of Maths Genie

Alternative Handouts

• Acute Learning Problem Pairs as PowerPoint or PDF

Online Platforms (with Videos & Exercises)

• Khan Academy
  • Solving Exponential Models
  • Exponential Models (Whole Set of Lessons)

Activities

• Berwick Maths Diagnostic Questions
• thechalkface
  • Exponential Rabbits
  • Carbon Dating
  • Over the Moon
  • Linear Laws Damping
• Math = Love (Sarah Carter)
  • Semi-log Plots activities
  • Semi-Log Plot Bingo
  • Semi-Log Graph Paper
  • Mystery Graphing Picture
• Don Steward
  • Diamonds Carat Size
  • Alkali Metals Power Relationship
• Ticktock Maths
  • Logarithms and non-linear data lesson (slides)

Worksheets

• Jethwa Maths Starter Activities (p13-16)
• Mathsaurus (Kevin Olding)
  • Linearising with logarithms (exam questions)
• MadAsMaths (T. Madas)
  • Logarithms Exam Questions Q64 (difficulty 4 of 5)
  • Linearization of Graphs Q1–5, 9–13, 18, 19, 21 (difficulty 2–5 of 5)

Whole Topic Videos

Exponentials & Logarithms 22 • Exam Questions • P1 Ex14 ExamQs • 🤖

Bicen Maths Exam Questions

External Whole Topic Past Paper Style Questions

• Jethwa Maths Worksheet & Topic Tests
• Dr Frost Topic Test✦ (Teacher Log in Required)
• naikermaths PPQ booklet✦
• PMT
  • Exponential Equations 1 · 2
  • Graphs of Exponentials and Logs
  • Laws of Logarithms 1 · 2
  • Cheat Sheet
• Transum
  • Exam-style questions: Logs

Activities

• Math = Love (Sarah Carter)
  • Earthquakes & Explosions logs task
  • Amy Gruen's Exponent puzzles
• MEI
  • MEI teaching notes, misconceptions & prompts: Exponentials and logarithms✦
• GeoGebra
  • Exponential growth vs linear growth✦
• Desmos
  • Match my exponential graph✦
• Andy Lutwyche
  • Codebreakers
  • Erica’s Errors
  • Codebreaker 5

Worksheets

• Mathshelper (Jon Stone)
  • Exponentials And Logarithms
• MathedUp
  • Exponentials & logs: Questions · Challenge · Answers
• Maths Genie
  • Exponentials and Logarithms: Questions · Solutions · Lesson · Topic page
• Jethwa Maths
  • Year 1 Exponentials and Logarithms: Worksheets · Unit Test
• Mathsaurus (Kevin Olding)
  • Logarithms (exam questions)
• MME
  • C2 Logarithms: Questions | Answers  •  C3 Exponentials & Natural Logarithms: Questions | Answers
• MadAsMaths (T. Madas)
  • Logarithms Exam Questions Q92, 96, 107–108, 113–114, 120 (difficulty 4–5 of 5)
• Dr Austin Maths
  • Revision Practice Grid (Word | PDF | Answers)
  • Fill in the Blanks (Word | PDF | Answers)
  • Solving Equations Crack the Code (Word | PDF | Answers)