Unit 03 · Lesson 03 · Week 7 · Exponential & Logarithmic Functions
Logarithms as Inverses and Exponents
Use logarithms as inverse functions and exponents to solve equations, compare scales, and linearize multiplicative relationships. In this lesson, the central focus is logarithms as inverses and exponents. Learners move among symbolic, graphical, numerical, and contextual evidence rather than relying on one representation alone.
Learning intentions
- Interpret the defining quantities and notation used in logarithms as inverses and exponents.
- Construct or analyze at least two representations related to logarithms as inverses and exponents.
- Show symbolic work with correct domains, units, intervals, or angle measures.
- Explain why the selected method is valid and check the result against another representation.
Essential question
How do multiplicative change, inverse structure, logarithmic scales, and parameter interpretation support defensible growth and decay models?
Multiple-representation model
Use logarithms as inverse functions and exponents to solve equations, compare scales, and linearize multiplicative relationships.
\log_b(x)=y\Longleftrightarrow b^y=x\log_b(MN)=\log_bM+\log_bN\log_bM=\frac{\ln M}{\ln b}
Worked reasoning
Misconception check: Logarithms distribute over products and quotients, not over sums and differences.
Communication and reasoning
Write a complete response that identifies the representation used, shows the symbolic steps, interprets parameters or units, and verifies the conclusion with a second representation or reasonableness check.
Khaemenes Academy · Open-Age Precalculus
Logarithms as Inverses and Exponents
Analyze, calculate, graph, or justify each result. Show reasoning, intervals, units, and calculator evidence when used.
Name
Date
Rewrite log₂32=5 in exponential form.
Evaluate log₁₀0.001.
Expand ln(3x²).
Condense 2lnx−ln(x+1).
Solve log₃(x−1)=2.
Use change of base to express log₇20.