g10-06-07 · Core topics
Products become sums of logarithms
Is logₐ(u+v)=logₐu+logₐv?
1 MIN · PREDICT ON PAPER
For base 2 and u=v=2, predict the logarithms of the product and of the sum.
Discuss or write first. No typed answer is required to continue.
Product rule: 2=2. Sum claim: 2 happens to equal 2; one accidental equality is not a general law.
2 MIN · EXPLAIN AND REVISE
Which condition does your claim need?
Compare at least two cases. Write a reason, then check the explanation together.
Reveal explanation and conditions
For positive u,v, logₐ(uv)=logₐu+logₐv. There is no corresponding sum rule. With u=2,v=4, the right side is 3 while log₂6 differs.
Conditions: Require a>0,a≠1 and u,v>0 separately. A positive product of two negative numbers does not make their separate real logarithms exist.
Mathematical justification: Write u=aʳ,v=aˢ. Then uv=a^(r+s), u/v=a^(r−s), and uᵏ=a^(kr). For valid base c, logₐu=log_cu/log_ca.
1 MIN · EXIT CHECK
Evaluate log₂8−log₂2 and explain the single-log form.
Check after answering
2, equal to log₂(8/2)=log₂4.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: First use u=v=2 to expose an accidental agreement; then change v to 4 to demand a true counterexample.
Watch for: A rule for the logarithm of a product also works for a sum.
Challenge: Use the product rule to derive quotient, power and change-of-base formulas, stating all domain conditions.
Suggested timing: predict 1 min, compare 3 min, explain 2 min, exit check 1 min. Observations can suggest a claim; they do not prove it for every input.
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