Logarithm Properties Today

Overview

In this sixth lesson of the unit, students connect logarithmic functions to their exponential inverses and then explore key graph features (intercepts and asymptote) to support solving logarithmic equations and interpreting parameter effects. This builds from earlier work on logarithms as tools for transforming equations and prepares for future lessons involving technology and contextual applications.

Learning intentions

Students will:

• identify logarithmic functions as the inverse of exponential functions and interpret what this means for inputs and outputs
• recognise and determine qualitative features of graphs of the form (y=\log_a(x)) for (a>1)
• explain how the parameter (a) affects the shape, and how (h) and (k) shift the graph for (y=\log_a(x-h)+k)
• solve simple logarithmic equations by selecting an appropriate strategy, with and without technology

Success criteria

Students can:

• state the domain and the vertical asymptote of (y=\log_a(x)) and locate the intercept(s) correctly
• describe how changing (a), (h), and (k) changes where the log graph sits and how steep it is
• solve a basic logarithmic equation and justify each step (e.g., using inverse logic and/or logarithm laws)
• check their solution by substituting back and confirming the expression is defined

Curriculum links

• Logarithmic functions: recognise and determine qualitative features of the graph of (y=\log_a(x)) including asymptote and intercept
• Logarithmic functions: describe the effect of parameters (a), (h), and (k) on (y=\log_a(x-h)+k)
• Logarithmic functions: solve equations involving logarithmic functions with and without technology
• (Foundational for today’s solving) Logarithmic laws and definitions to simplify expressions

Lesson structure (60 minutes)

1. 0–6 min · Retrieval and hook. Teacher writes an exponential function and its inverse relation on the board: “If (y=a^x), then (x=\log_a(y)).” Students complete a quick check: convert two given exponent statements into log statements and vice versa.

2. 6–16 min · Direct teach: inverse meaning. Teacher guides discussion: explain how swapping (x) and (y) for an inverse affects function behaviour and the graph; then demonstrate the correspondence using a small table (e.g., for (a=2)). Students use a table to generate points for (y=\log_2(x)) and then sketch a rough curve.

3. 16–26 min · Graph features: (y=\log_a(x)). Teacher explicitly focuses on qualitative features: domain (x>0), vertical asymptote, and intercept(s) for (a>1). Students answer two teacher prompts:

• “Where is the vertical asymptote and why?”
• “What point does the graph cross when (x=1), and what is (y) there?”

4. 26–38 min · Parameter effects: (y=\log_a(x-h)+k). Teacher models transformations in a stepwise way: inner shift ((x-h)) and outer shift (+k), then the role of base (a) on steepness. Students complete a structured “compare and predict” task: given two equations, they state the new vertical asymptote and one shifted point (e.g., where the inside becomes 1: (x-h=1)).

5. 38–52 min · Solving logarithmic equations (no tech first). Teacher selects a short equation where inverse logic works cleanly, such as:

• Solve (\log_3(x)=2) and then solve a second one like (\log_3(x)=\log_3(9)) (students choose whether to use laws or inverse). Students solve without technology, showing justification, then check by substitution and verifying (x) is in the domain.

6. 52–57 min · Technology check and refinement. Teacher demonstrates how a graphing calculator or maths tool can confirm solutions and highlight invalid values. Students graph the function(s) or use solver features to verify at least one solution from the previous step.

7. 57–60 min · Exit ticket. Teacher collects a quick written response:

• For (y=\log_{4}(x-3)+2), state the vertical asymptote and one x-intercept candidate or an easily found point.
• Then solve (\log_{4}(x)=\tfrac{1}{2}) and state the domain condition.

Resources

• Board/whiteboard and markers
• Graph paper or printed sketch grids
• Student device with graphing/calculation capability (optional but recommended for verification step)
• Teacher-prepared equation cards (2–4 short logarithmic equations)
• Parameter worksheets with “asymptote + point” prompts
• Exit ticket slips

Assessment

• Formative: teacher listens during inverse/table building to confirm students link (a^x) and (\log_a(x))
• Formative: check students’ qualitative feature answers (asymptote, intercepts, predicted shifts) during steps 3–4
• Summative-in-miniature via exit ticket: accuracy of asymptote/point identification and correct equation solving with domain checking

Differentiation

• Support: provide sentence starters such as “Because the log argument must be positive, (x- h > 0), so …” and “When (x-h=1), (\log_a(1)=0), so the point is …”
• Support: give one worked example of parameter identification (asymptote and a key point) before releasing a second equation to students
• Extension: for students who are confident, ask them to explain how steepness changes when (a) increases (from 2 to 5) using a brief comparison of graphs or tables
• EAL/SEN: allow use of symbols and minimal prose; provide a template for solutions: “Step → reason → check domain → substitution check”