BetterGrades Precalculus · Unit 7 · Lesson
Logarithm laws
Derive and apply product, quotient, and power laws while preserving positive-argument domains.
Start with the situation
Logarithm laws convert products to sums, quotients to differences, and powers to coefficients.
Multiplicative models describe repeated percentage change, while logarithms recover the time or exponent hidden inside that process. Together they support growth, decay, finance, regression, and bounded models.
Prerequisite check
- Use exponent laws.
- Interpret function parameters.
- Distinguish exact and approximate values.
Explanation
Factor multiplicative structure, apply the laws, and preserve positive-argument conditions.
No law separates a logarithm of a sum or difference.
A symbolic answer is not complete by itself. In this lesson, the same claim must also be readable through exponent-to-log derivation, valid-invalid law gallery, or another equivalent representation.
What the idea is really doing
Exponential change multiplies over equal input steps, while logarithms answer the inverse question: what exponent produces a given output? Parameters must be interpreted as an initial value, a multiplier, a rate, or a long-run bound—not as decoration.
This lesson narrows that lens to one goal: derive and apply product, quotient, and power laws while preserving positive-argument domains. The point is not to memorize an isolated trick; it is to know what evidence makes the conclusion valid and how a second representation can check it.
Plan before calculating
Problem
Expand
- Plan
- Start by identifying the mathematical structure in the prompt. Then use the lesson method rather than guessing from appearance: Factor multiplicative structure, apply the laws, and preserve positive-argument conditions.
- Conclusion
- Why the check works
- Product, power, and quotient laws combine.
See the idea in three forms
foundation example
Expand
Solution
Product, power, and quotient laws combine.
representation example
Expand
Solutionlog_bx-log_by.
This example expresses logarithm laws in a second form.
transfer example
Condense .
Solution
No law separates a logarithm of a sum or difference.
Read this graph as text
Logarithm laws · Exponent-to-log derivation. Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Product, power, and quotient laws combine. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Derive and apply product, quotient, and power laws while preserving positive-argument domains.
Follow the foundation example from its given information to the conclusion. The labels identify the mathematical feature that makes the result valid: Product, power, and quotient laws combine.
Read this graph as text
Logarithm laws · Valid-invalid law gallery. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for logarithm laws. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Derive and apply product, quotient, and power laws while preserving positive-argument domains.
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for logarithm laws.
Read this graph as text
Logarithm laws · Expansion-condensation tree. Compare the valid path with the tempting shortcut. The figure shows why writing log(x+y)=logx+logy leads to a false conclusion. The figure uses concrete points, curves, arrows, intervals, or matrix structure instead of relying on color alone.
Labels, point shapes, line styles, arrows, and position carry the mathematical meaning; color is supplementary.
Why it matters: Use the mathematical objects in this figure to support the lesson outcome: Derive and apply product, quotient, and power laws while preserving positive-argument domains.
Compare the valid path with the tempting shortcut. The figure shows why writing leads to a false conclusion.
Find the first invalid move
A frequent error is writing .
Expand log_b(xy).
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10 concrete questions
01Expand log_b(xy).
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02Expand
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03Condense .
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04Is ?
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05Explain why this conclusion is valid: . Use the foundation problem as evidence: Expand .
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06Solve the representation example, then name the feature of logarithm laws that it illustrates: Expand
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07Correct this reasoning and identify the first unsafe assumption: A frequent error is writing .
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08Connect two representations for this example: Expand . Describe what a graph, table, mapping, or algebraic form would have to show.
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09Create a nearby example by changing one number or condition in this prompt: Condense . Predict the effect, solve your new example, and compare it with the original.
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10Write a short verification checklist for logarithm laws, then apply it to one worked example from this lesson.
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Connect forward
The next lesson, Solving exponential equations, uses this result as part of a larger structure.
Source record
Original BetterGrades manuscript, rights-separated references.
- Yoshiwara, Modeling, Functions, and Graphs
- Lippman and Rasmussen, Precalculus Volume 1
- Stitz and Zeager, Precalculus
No long source passage is reproduced.