Mathematics · Grade 11 · Algebra 2

Logarithms Laws Domains and Valid Equations

Interpret a logarithm as the inverse of an exponential rule. Apply product laws and solve with positive-argument restrictions. Explain invalid laws, extraneous roots and base conditions.

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Learning goals

Interpret a logarithm as the inverse of an exponential rule.
Apply product laws and solve with positive-argument restrictions.
Explain invalid laws, extraneous roots and base conditions.

Before you begin

Use positive-base powers, inverse functions, factoring and original-equation domain checks.

Age and grade do not measure readiness. Adjust reading or adult support to preserve the intended learning goal.

Follow the sequence

Complete practice

Logarithms Laws Domains and Valid Equations

Logarithms reverse positive powers

Use base 2. For positive y, log_2(y) = x means 2^x = y. Consider log_2(x − 1) + log_2(x + 1) = 3. Both logarithm arguments must be positive. Apply a law only on its valid domain.

Worked example: log₄(16)=2 because 4²=16. For positive u and v, log₄(u)+log₄(v)=log₄(uv); adding the arguments is a different operation. Use the material to work through each task. Show the requested response.

1.Evaluate log_2(8).

Practice preview

Grade 11

Key

Logarithms Laws Domains and Valid Equations

Practice · Independent · 45 min

Interpret a logarithm as the inverse of an exponential rule. Apply product laws and solve with positive-argument restrictions. Explain invalid laws, extraneous roots and base conditions. A complete set with exploration, practice and application. Use the linked unit for the learning sequence and teaching guide.

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1. Explore

Logarithms Laws Domains and Valid Equations: Explore and Explain

Logarithms reverse positive powers

Use base 2. For positive y, log_2(y) = x means 2^x = y. Consider log_2(x − 1) + log_2(x + 1) = 3. Both logarithm arguments must be positive. Apply a law only on its valid domain.

Worked example: log₄(16)=2 because 4²=16. For positive u and v, log₄(u)+log₄(v)=log₄(uv); adding the arguments is a different operation. Explore the material. Talk through your reasoning with suitable support.

1.Evaluate log_2(8).

Practice preview

Grade 11

Key

Logarithms Laws Domains and Valid Equations: Explore and Explain

Explore · Independent · 12 min

Interpret a logarithm as the inverse of an exponential rule. Apply product laws and solve with positive-argument restrictions. Explain invalid laws, extraneous roots and base conditions. A focused learn activity. Use the linked unit for the learning sequence and teaching guide.

Choose another practice set

Preview / Print →

2. Practice

Logarithms Laws Domains and Valid Equations: Practice Independently

Logarithms reverse positive powers

Use base 2. For positive y, log_2(y) = x means 2^x = y. Consider log_2(x − 1) + log_2(x + 1) = 3. Both logarithm arguments must be positive. Apply a law only on its valid domain.

Use the material to work through each task. Show the requested response.

1.State the real domain for the two logarithms in the material.

Practice preview

Grade 11

Key

Logarithms Laws Domains and Valid Equations: Practice Independently

Practice · Independent · 20 min

Interpret a logarithm as the inverse of an exponential rule. Apply product laws and solve with positive-argument restrictions. Explain invalid laws, extraneous roots and base conditions. A focused practice activity. Use the linked unit for the learning sequence and teaching guide.

Choose another practice set

Preview / Print →

3. Apply

Logarithms Laws Domains and Valid Equations: Apply and Create

Logarithms reverse positive powers

Use base 2. For positive y, log_2(y) = x means 2^x = y. Consider log_2(x − 1) + log_2(x + 1) = 3. Both logarithm arguments must be positive. Apply a law only on its valid domain.

Use what you learned to explain, construct or apply an idea.

1.A learner writes log_2(u) + log_2(v) = log_2(u + v). Correct this for positive u and v. Check using u = v = 2.

Practice preview

Grade 11

Key

Logarithms Laws Domains and Valid Equations: Apply and Create

Apply · Independent · 25 min

Interpret a logarithm as the inverse of an exponential rule. Apply product laws and solve with positive-argument restrictions. Explain invalid laws, extraneous roots and base conditions. A focused apply activity. Use the linked unit for the learning sequence and teaching guide.

Choose another practice set

Preview / Print →

Teaching and response guide

Begin with: Evaluate log_2(8).
Connect the response to: State the real domain for the two logarithms in the material.
Ask students to apply the idea: A learner writes log_2(u) + log_2(v) = log_2(u + v). Correct this for positive u and v. Check using u = v = 2.

Watch for: A condensed logarithm can hide invalid original arguments; base 1 cannot define a logarithm.

Check the answer criteria for each task. A supported explanation may have more than one acceptable response. These activities offer evidence to discuss; completing a sheet alone does not establish mastery.