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Grade 11 · lesson 3 of 4

Convert to Exponential Form and Check the Domain

A logarithmic equation can be rewritten using the definition of a logarithm.

About 25 minutes, plus any extended practice

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Supplied materials

  • Paper or a browser scratch area; a calculator is optional unless rounding is requested.
  • Supplied teaching case: Solve log₃(x−2)=2 over the reals. Candidate answers: (A) 10 (B) 11 (C) 12 (D) 22.
  • Independent check inputs: Solve log₃(x−3)=2 over the reals. Candidate answers: (A) 11 (B) 12 (C) 13 (D) 24.

Learn the idea

Logarithms reverse exponentiation with a valid positive base other than one. They answer exponent questions, reorganize multiplicative structure and solve exponential relationships while requiring positive real logarithm arguments. A logarithmic equation can be rewritten using the definition of a logarithm. Solving the resulting exponential relationship finds a candidate that must leave the original argument positive.

Here is the method on the supplied teaching case. Convert to exponential form: x−2=3²=9, so x=11. This satisfies the required x−2>0. A logarithm does not mean multiplication by its base.

Do not drop the shift inside the logarithm’s argument. Check that the final input satisfies the positive-argument condition; algebraic rearrangement alone cannot validate an undefined original logarithm.

  1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
  2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
  3. Solve the independent check with its changed inputs before revealing feedback.
  4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.
Worked example

See the method in action

Teaching case: Solve log₃(x−2)=2 over the reals. Candidate answers: (A) 10 (B) 11 (C) 12 (D) 22. Solution: Convert to exponential form: x−2=3²=9, so x=11. This satisfies the required x−2>0. A logarithm does not mean multiplication by its base. The final answer is 11. Do not drop the shift inside the logarithm’s argument. Check that the final input satisfies the positive-argument condition; algebraic rearrangement alone cannot validate an undefined original logarithm.

Your turn

Check your reasoning

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Helpful tips
  • A logarithmic equation can be rewritten using the definition of a logarithm.
  • Do not drop the shift inside the logarithm’s argument.
Solve log₃(x−3)=2 over the reals.

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See the answer and explanation

12

Convert to exponential form: x−3=3²=9, so x=12. This satisfies the required x−3>0. A logarithm does not mean multiplication by its base.

Take it further

Changed-input task: Solve log₃(x−9)=2 over the reals. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. Show the mathematical reason, not only the selected answer. Expected result: 18. Convert to exponential form: x−9=3²=9, so x=18. This satisfies the required x−9>0. A logarithm does not mean multiplication by its base.

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More practice: try new cases

Choose an answer for each problem. Explain your method on paper, then use the feedback to decide what to revisit.

More practice: try new cases
1. Solve log₃(x−4)=2 over the reals.
Need a hint?
  • A logarithmic equation can be rewritten using the definition of a logarithm.
  • Do not drop the shift inside the logarithm’s argument.
2. Solve log₃(x−5)=2 over the reals.
Need a hint?
  • A logarithmic equation can be rewritten using the definition of a logarithm.
  • Do not drop the shift inside the logarithm’s argument.
3. Solve log₃(x−6)=2 over the reals.
Need a hint?
  • A logarithmic equation can be rewritten using the definition of a logarithm.
  • Do not drop the shift inside the logarithm’s argument.

Your choices are temporary. Feedback appears after you check all the answers.

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