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M3-030-R03-V04
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F-LE.4.1
Core
M3-030-R03-V04
Prove simple logarithm laws.
Derive and domain-check the logarithm power law
Problem
Audit \(\log_{3}((-2)^2)~=~2\log_{3}(-2)\) over the real numbers. State the definedness of each side, whether they are equivalent, and the power-law argument requirement.
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A
Candidate analysis: The left side is log₃(4), which is defined, but log₃(−2) on the right is undefined over the reals. The expressions are not equivalent; the power-law rewrite requires a positive original argument.
B
Candidate analysis: The left side is log₃(4), which is defined, but log₃(−2) on the right is undefined over the reals. The expressions are equivalent; the power-law rewrite requires a positive original argument.
C
Candidate analysis: The left side is log₃(4), which is defined, but log₃(−2) on the right is defined over the reals. The expressions are not equivalent; the power-law rewrite requires a positive original argument.
D
Candidate analysis: The left side is log₃(4), which is defined, but log₃(−2) on the right is undefined over the reals. The expressions are not equivalent; the power-law rewrite allows any nonzero original argument.
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Curriculum context
Standard F-LE.4.1
Category Functions
Domain Linear, Quadratic, and Exponential Models
Objective Prove simple logarithm laws.
Problem type Derive and domain-check the logarithm power law
