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