g10-06-05 · Core topics
A logarithm asks for an exponent
What number does log₂8 ask for, and why is log₂0 impossible?
1 MIN · PREDICT ON PAPER
Complete 2^?=8 and 2^?=1/4 before selecting logarithm inputs.
Discuss or write first. No typed answer is required to continue.
2^3 = 8, so log_2(8)=3.
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
logₐb is the exponent y satisfying aʸ=b. Thus log₂8=3 and log₂(1/4)=−2. A positive exponential never equals zero or a negative number.
Conditions: A real logarithm requires a>0, a≠1 and argument b>0. A negative logarithm value is allowed.
Mathematical justification: The exponential aʸ is positive and strictly monotone for a>0,a≠1; it therefore has exactly one exponent for each positive output.
1 MIN · EXIT CHECK
Evaluate log₃(1/9).
Check after answering
−2 because 3⁻²=1/9.
Teacher guidance and extension
Before: Real-number operations and reading coordinates; write a prediction on paper.
Ask: Always read logₐb aloud as “a to what exponent equals b?”.
Watch for: The logarithm value must be positive; logₐ0=0.
Challenge: Explain why base 1 gives either no exponent or infinitely many, not a logarithm function.
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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