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Grade 11 · unit review

Review: Algebra2: Logarithms and Exponential Equations

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.

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Unit review questions
1. Evaluate log₂(128).
Need a hint?
  • A logarithm log_b(A) asks which exponent on b yields A.
  • The logarithm is not division of the argument by the base.
2. For positive x,y, rewrite log_8(xy).
Need a hint?
  • The logarithm product rule reflects the addition of exponents when same-base powers multiply.
  • The rule does not split a sum inside a logarithm.
3. Solve log₃(x−7)=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.
4. A model is A(t)=7·2^t. At what t does A(t)=56?
Need a hint?
  • An exponential model can be solved by first dividing out the initial amount.
  • The period count describes the supplied model rather than guaranteed real growth.

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Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. Evaluate log₂(512). Candidate answers: (A) 10 (B) 18 (C) 8 (D) 9. 2. For positive x,y, rewrite log_10(xy). Candidate answers: (A) (log_10x)(log_10y) (B) log_10(x+y) (C) log_10x+log_10y (D) log_10x−log_10y. 3. Solve log₃(x−9)=2 over the reals. Candidate answers: (A) 17 (B) 18 (C) 19 (D) 36. 4. A model is A(t)=9·2^t. At what t does A(t)=72? Candidate answers: (A) 2 periods (B) 3 periods (C) 4 periods (D) 8 periods.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 9. A logarithm answers which exponent produces the argument. Since 2^9=512, log₂(512)=9. The argument is positive and the base is valid. 2. log_10x+log_10y. The product rule turns multiplication inside a logarithm into addition outside. It applies with positive factors and valid base 10; it does not split a sum or multiply logarithm values. 3. 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. 4. 3 periods. Divide by 9: 2^t=8=2³. Equal positive-base powers give t=3. A logarithm could solve the same equation when the target is not an exact power.

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