Logarithms

Logarithms

Logarithms

publish date

May 29, 2025

duration

Difficulty

Beginner

what you'll learn

Lesson details

Units 1 & 2 (Foundation of Logarithms)
Introduce the logarithmic function as the inverse of exponentiation, develop fluency with logarithmic laws, and apply these in solving exponential equations, including those with real-world modelling relevance (e.g. population growth, radioactive decay). Emphasize transformation and graphing of logarithmic and exponential curves, domain and range analysis, and use of technology for verification and exploration.

Unit 2 (Logarithmic Equations & Applications)
Extend logarithmic manipulation to equations involving multiple terms and variable exponents, apply logarithmic models to solve real-world problems in finance and science, and prepare algebraic skills essential for calculus contexts, such as simplifying complex expressions and solving equations analytically and graphically.

Unit 1: Foundations of Logarithmic and Exponential Relationships

1.1 Introduction to Logarithms and Exponentials

  • Exponential Laws Recap: index laws, base rules

  • Definition of Logarithms: log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = xlogb​(x)=y⟺by=x

  • Graphing Basics: exponential growth and decay

1.2 Logarithmic Laws and Manipulation

  • Logarithmic Properties:

    • Product Rule: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)logb​(xy)=logb​(x)+logb​(y)

    • Quotient Rule: log⁡b(x/y)=log⁡b(x)−log⁡b(y)\log_b(x/y) = \log_b(x) - \log_b(y)logb​(x/y)=logb​(x)−logb​(y)

    • Power Rule: log⁡b(xn)=nlog⁡b(x)\log_b(x^n) = n \log_b(x)logb​(xn)=nlogb​(x)

  • Change of Base Formula: log⁡b(x)=log⁡k(x)log⁡k(b)\log_b(x) = \frac{\log_k(x)}{\log_k(b)}logb​(x)=logk​(b)logk​(x)​

  • Simplifying and Expanding Logarithmic Expressions

1.3 Graphs and Transformations of Exponentials and Logarithms

  • Sketching: base eee, base 10, and base 2 functions

  • Domain, Range, and Asymptotes

  • Transformations: shifts, stretches, and reflections

  • Technology Integration: graphing tools for verification

1.4 Modelling with Exponentials

  • Real-World Contexts: population models, cooling/heating, finance (compound interest)

  • Interpreting Parameters: growth/decay rates, initial value

  • Comparing linear, exponential, and logarithmic growth

Unit 2: Solving and Applying Logarithmic Equations

2.1 Solving Logarithmic Equations

  • Equating Logs: log⁡b(x)=log⁡b(y)⇒x=y\log_b(x) = \log_b(y) \Rightarrow x = ylogb​(x)=logb​(y)⇒x=y

  • Using Log Laws to Simplify Before Solving

  • Mixed Equations: exponentials reducible via logs

  • Graphical Solutions: technology for approximate solutions

2.2 Real-World Applications of Logarithms

  • Financial Models: compound interest, depreciation

  • Scientific Models: pH scale, Richter scale, sound intensity

  • Inverse Interpretation: determining time or rate from log equations

2.3 Preparation for Calculus Concepts

  • Growth vs. Rate of Growth: interpretation in log/exponential form

  • Logarithmic Differentiation Preview (qualitative only)

  • Understanding asymptotic behavior in function growth

Units 1 & 2 (Foundation of Logarithms)
Introduce the logarithmic function as the inverse of exponentiation, develop fluency with logarithmic laws, and apply these in solving exponential equations, including those with real-world modelling relevance (e.g. population growth, radioactive decay). Emphasize transformation and graphing of logarithmic and exponential curves, domain and range analysis, and use of technology for verification and exploration.

Unit 2 (Logarithmic Equations & Applications)
Extend logarithmic manipulation to equations involving multiple terms and variable exponents, apply logarithmic models to solve real-world problems in finance and science, and prepare algebraic skills essential for calculus contexts, such as simplifying complex expressions and solving equations analytically and graphically.

Unit 1: Foundations of Logarithmic and Exponential Relationships

1.1 Introduction to Logarithms and Exponentials

  • Exponential Laws Recap: index laws, base rules

  • Definition of Logarithms: log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = xlogb​(x)=y⟺by=x

  • Graphing Basics: exponential growth and decay

1.2 Logarithmic Laws and Manipulation

  • Logarithmic Properties:

    • Product Rule: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)logb​(xy)=logb​(x)+logb​(y)

    • Quotient Rule: log⁡b(x/y)=log⁡b(x)−log⁡b(y)\log_b(x/y) = \log_b(x) - \log_b(y)logb​(x/y)=logb​(x)−logb​(y)

    • Power Rule: log⁡b(xn)=nlog⁡b(x)\log_b(x^n) = n \log_b(x)logb​(xn)=nlogb​(x)

  • Change of Base Formula: log⁡b(x)=log⁡k(x)log⁡k(b)\log_b(x) = \frac{\log_k(x)}{\log_k(b)}logb​(x)=logk​(b)logk​(x)​

  • Simplifying and Expanding Logarithmic Expressions

1.3 Graphs and Transformations of Exponentials and Logarithms

  • Sketching: base eee, base 10, and base 2 functions

  • Domain, Range, and Asymptotes

  • Transformations: shifts, stretches, and reflections

  • Technology Integration: graphing tools for verification

1.4 Modelling with Exponentials

  • Real-World Contexts: population models, cooling/heating, finance (compound interest)

  • Interpreting Parameters: growth/decay rates, initial value

  • Comparing linear, exponential, and logarithmic growth

Unit 2: Solving and Applying Logarithmic Equations

2.1 Solving Logarithmic Equations

  • Equating Logs: log⁡b(x)=log⁡b(y)⇒x=y\log_b(x) = \log_b(y) \Rightarrow x = ylogb​(x)=logb​(y)⇒x=y

  • Using Log Laws to Simplify Before Solving

  • Mixed Equations: exponentials reducible via logs

  • Graphical Solutions: technology for approximate solutions

2.2 Real-World Applications of Logarithms

  • Financial Models: compound interest, depreciation

  • Scientific Models: pH scale, Richter scale, sound intensity

  • Inverse Interpretation: determining time or rate from log equations

2.3 Preparation for Calculus Concepts

  • Growth vs. Rate of Growth: interpretation in log/exponential form

  • Logarithmic Differentiation Preview (qualitative only)

  • Understanding asymptotic behavior in function growth

Units 1 & 2 (Foundation of Logarithms)
Introduce the logarithmic function as the inverse of exponentiation, develop fluency with logarithmic laws, and apply these in solving exponential equations, including those with real-world modelling relevance (e.g. population growth, radioactive decay). Emphasize transformation and graphing of logarithmic and exponential curves, domain and range analysis, and use of technology for verification and exploration.

Unit 2 (Logarithmic Equations & Applications)
Extend logarithmic manipulation to equations involving multiple terms and variable exponents, apply logarithmic models to solve real-world problems in finance and science, and prepare algebraic skills essential for calculus contexts, such as simplifying complex expressions and solving equations analytically and graphically.

Unit 1: Foundations of Logarithmic and Exponential Relationships

1.1 Introduction to Logarithms and Exponentials

  • Exponential Laws Recap: index laws, base rules

  • Definition of Logarithms: log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = xlogb​(x)=y⟺by=x

  • Graphing Basics: exponential growth and decay

1.2 Logarithmic Laws and Manipulation

  • Logarithmic Properties:

    • Product Rule: log⁡b(xy)=log⁡b(x)+log⁡b(y)\log_b(xy) = \log_b(x) + \log_b(y)logb​(xy)=logb​(x)+logb​(y)

    • Quotient Rule: log⁡b(x/y)=log⁡b(x)−log⁡b(y)\log_b(x/y) = \log_b(x) - \log_b(y)logb​(x/y)=logb​(x)−logb​(y)

    • Power Rule: log⁡b(xn)=nlog⁡b(x)\log_b(x^n) = n \log_b(x)logb​(xn)=nlogb​(x)

  • Change of Base Formula: log⁡b(x)=log⁡k(x)log⁡k(b)\log_b(x) = \frac{\log_k(x)}{\log_k(b)}logb​(x)=logk​(b)logk​(x)​

  • Simplifying and Expanding Logarithmic Expressions

1.3 Graphs and Transformations of Exponentials and Logarithms

  • Sketching: base eee, base 10, and base 2 functions

  • Domain, Range, and Asymptotes

  • Transformations: shifts, stretches, and reflections

  • Technology Integration: graphing tools for verification

1.4 Modelling with Exponentials

  • Real-World Contexts: population models, cooling/heating, finance (compound interest)

  • Interpreting Parameters: growth/decay rates, initial value

  • Comparing linear, exponential, and logarithmic growth

Unit 2: Solving and Applying Logarithmic Equations

2.1 Solving Logarithmic Equations

  • Equating Logs: log⁡b(x)=log⁡b(y)⇒x=y\log_b(x) = \log_b(y) \Rightarrow x = ylogb​(x)=logb​(y)⇒x=y

  • Using Log Laws to Simplify Before Solving

  • Mixed Equations: exponentials reducible via logs

  • Graphical Solutions: technology for approximate solutions

2.2 Real-World Applications of Logarithms

  • Financial Models: compound interest, depreciation

  • Scientific Models: pH scale, Richter scale, sound intensity

  • Inverse Interpretation: determining time or rate from log equations

2.3 Preparation for Calculus Concepts

  • Growth vs. Rate of Growth: interpretation in log/exponential form

  • Logarithmic Differentiation Preview (qualitative only)

  • Understanding asymptotic behavior in function growth

About Author

Methods Tutor

Methods Tutor

Methods Tutor

Theo – Tutor Summary

  • Specialty: Mathematical Methods

  • ATAR: 99.65

  • School: Xavier College Alumni

  • Subject Achievements:


    • 50 Raw in Global Politics

    • 47 Raw in History Revolutions

    • 47 Raw in Mathematical Methods

    • 44 Raw in English

Theo – Tutor Summary

  • Specialty: Mathematical Methods

  • ATAR: 99.65

  • School: Xavier College Alumni

  • Subject Achievements:


    • 50 Raw in Global Politics

    • 47 Raw in History Revolutions

    • 47 Raw in Mathematical Methods

    • 44 Raw in English

Theo – Tutor Summary

  • Specialty: Mathematical Methods

  • ATAR: 99.65

  • School: Xavier College Alumni

  • Subject Achievements:


    • 50 Raw in Global Politics

    • 47 Raw in History Revolutions

    • 47 Raw in Mathematical Methods

    • 44 Raw in English

Up next…

Up next…