Logarithms
Logarithms
Logarithms
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: logb(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: logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb(y)
Quotient Rule: logb(x/y)=logb(x)−logb(y)\log_b(x/y) = \log_b(x) - \log_b(y)logb(x/y)=logb(x)−logb(y)
Power Rule: logb(xn)=nlogb(x)\log_b(x^n) = n \log_b(x)logb(xn)=nlogb(x)
Change of Base Formula: logb(x)=logk(x)logk(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: logb(x)=logb(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: logb(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: logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb(y)
Quotient Rule: logb(x/y)=logb(x)−logb(y)\log_b(x/y) = \log_b(x) - \log_b(y)logb(x/y)=logb(x)−logb(y)
Power Rule: logb(xn)=nlogb(x)\log_b(x^n) = n \log_b(x)logb(xn)=nlogb(x)
Change of Base Formula: logb(x)=logk(x)logk(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: logb(x)=logb(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: logb(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: logb(xy)=logb(x)+logb(y)\log_b(xy) = \log_b(x) + \log_b(y)logb(xy)=logb(x)+logb(y)
Quotient Rule: logb(x/y)=logb(x)−logb(y)\log_b(x/y) = \log_b(x) - \log_b(y)logb(x/y)=logb(x)−logb(y)
Power Rule: logb(xn)=nlogb(x)\log_b(x^n) = n \log_b(x)logb(xn)=nlogb(x)
Change of Base Formula: logb(x)=logk(x)logk(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: logb(x)=logb(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
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