Mathematics · Grade 12

Active learning ideas

Solving Logarithmic Equations

Active learning works well for solving logarithmic equations because students often struggle with keeping track of domain restrictions while manipulating terms. Moving around the room, discussing with peers, and checking their own work helps them internalize the importance of valid arguments in logarithms.

• Ontario Curriculum Expectations
• HSA.REI.D.11

• 25–45 min
• Pairs → Whole Class
• 4 activities

Activity 01

• Collaborative Problem-Solving
• 35 min · Pairs

Pairs Relay: Log Solving Challenge

Pair students at whiteboards with a set of 6 log equations. One student solves while the partner times and checks domain; switch roles after each equation. Circulate to prompt property use, then class debriefs patterns.

Analyze why it is necessary to check for extraneous solutions when solving logarithmic equations.

Facilitation Tip

During the Pairs Relay, stand at the back of the room to monitor pacing and ensure students explain each step aloud to their partner.

What to look for

Present students with the equation log₂(x) + log₂(x-2) = 3. Ask them to identify the domain restrictions for each logarithm and then solve the equation, clearly indicating any extraneous solutions.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 02

• Collaborative Problem-Solving
• 45 min · Small Groups

Small Groups: Extraneous Error Stations

Set up 4 stations with pre-solved log equations, half containing extraneous solutions. Groups rotate, identify invalids, explain domains, and rewrite correctly. Share one insight per group.

Construct a strategy for solving logarithmic equations that involve multiple logarithmic terms.

Facilitation Tip

For Extraneous Error Stations, prepare a timer visible to all groups to keep rotations smooth and discussions focused.

What to look for

Provide students with a solved logarithmic equation that contains an extraneous solution. Ask them to write a brief explanation, in their own words, detailing why the extraneous solution is invalid and how it could have been avoided.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 03

• Collaborative Problem-Solving
• 25 min · Whole Class

Whole Class: Graph Match Verification

Display 5 log equations; class predicts valid solutions via thumbs up/down. Graph both sides on shared Desmos screen to confirm domains and intersections. Discuss mismatches.

Evaluate the validity of solutions to logarithmic equations based on their domain.

Facilitation Tip

In Graph Match Verification, circulate with colored pens to mark student graphs and prompt immediate corrections where needed.

What to look for

Pose the question: 'When solving log_b(argument) = c, why is it sufficient to check if the argument is positive, but when solving log_b(argument1) = log_b(argument2), you must check if both argument1 and argument2 are positive?' Facilitate a class discussion on the nuances of domain checking.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

Activity 04

• Collaborative Problem-Solving
• 30 min · Individual

Individual: Custom Equation Builder

Students craft a log equation with one extraneous solution, solve it themselves first. Exchange with a partner for independent solving and checking. Regroup to showcase.

Analyze why it is necessary to check for extraneous solutions when solving logarithmic equations.

What to look for

Present students with the equation log₂(x) + log₂(x-2) = 3. Ask them to identify the domain restrictions for each logarithm and then solve the equation, clearly indicating any extraneous solutions.

• Apply
• Analyze
• Evaluate
• Create
• Relationship Skills
• Decision-Making
• Self-Management

A few notes on teaching this unit

Start with a quick review of domain rules, then model solving one equation step-by-step while thinking aloud about checking arguments. Use color-coding on the board to highlight positive regions for arguments, as visual cues help students retain rules. Avoid rushing to combine logs; emphasize verifying each argument before and after combining terms to reinforce habits.

By the end of these activities, students will confidently solve logarithmic equations while consistently verifying solutions against domain rules. They will also identify and explain why extraneous solutions occur, using both algebraic and graphical reasoning.

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Watch Out for These Misconceptions

• During Pairs Relay, watch for students who solve equations quickly but skip checking if arguments are positive.

  Remind pairs to pause after each solution and ask, 'Is the argument positive?' before moving to the next step. If not, they should backtrack to find a valid solution.

• During Extraneous Error Stations, watch for students who assume combined logs automatically satisfy domain rules.

  Have groups sort their solutions into 'Valid' and 'Invalid' columns, then write the original arguments for each log to confirm positivity before combining.

• During Graph Match Verification, watch for students who only check the final graph and ignore intermediate steps.

  Ask students to label each graph with its corresponding equation and domain, then compare solutions visually to see where arguments become non-positive.

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Methods used in this brief

• Collaborative Problem-Solving