BetterGrades Precalculus · Unit 7 · Lesson

Logarithm laws

Derive and apply product, quotient, and power laws while preserving positive-argument domains.

Opening

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.

Before you begin

Prerequisite check

Use exponent laws.
Interpret function parameters.
Distinguish exact and approximate values.

Core explanation

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.

Conceptual reading

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.

Reusable method

A reliable route through the problem

Factor multiplicative structure.
Apply the laws.
Preserve positive-argument conditions.

Verification: Test the model at input zero and one step later, confirm the multiplier or inverse relationship, and state whether the domain and long-run behavior make sense in context.

Foundation walkthrough

Plan before calculating

Problem

Expand $log_b(\frac{5x^3}{y})$

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: $log_b5+3log_bx-log_by$
Why the check works: Product, power, and quotient laws combine.

Worked examples

See the idea in three forms

foundation example

Expand $log_b(\frac{5x^3}{y})$

Solution $log_b5+3log_bx-log_by$

Product, power, and quotient laws combine.

representation example

Expand $log_b(\frac{x}{y})$

Solutionlog_bx-log_by.

This example expresses logarithm laws in a second form.

transfer example

Condense $3lnx-lny$ .

Solution $ln(\frac{x^3}{y})$

No law separates a logarithm of a sum or difference.

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. — **Anchor figure · 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.

Valid-invalid law gallery. Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for logarithm laws. — **Mechanism figure · Valid-invalid law gallery**
Read the numbered reasoning path in order. Each stage preserves the quantities, restrictions, or structural conditions needed for 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. — **Comparison and error figure · 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.

Common mistake

Find the first invalid move

A frequent error is writing $log(x+y)=logx+logy$ .

Check yourself

Expand log_b(xy).

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice

Ten concrete questions

Practice 101

Expand log_b(xy).

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 202

Expand $log_b(\frac{x}{y})$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 303

Condense $3lnx-lny$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 404

Is $log(x+2)=logx+log2$ ?

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 505

Explain why this conclusion is valid: $log_b5+3log_bx-log_by$ . Use the foundation problem as evidence: Expand $log_b(\frac{5x^3}{y})$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 606

Solve the representation example, then name the feature of logarithm laws that it illustrates: Expand $log_b(\frac{x}{y})$

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 707

Correct this reasoning and identify the first unsafe assumption: A frequent error is writing $log(x+y)=logx+logy$ .

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 808

Connect two representations for this example: Expand $log_b(\frac{5x^3}{y})$ . Describe what a graph, table, mapping, or algebraic form would have to show.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 909

Create a nearby example by changing one number or condition in this prompt: Condense $3lnx-lny$ . Predict the effect, solve your new example, and compare it with the original.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Practice 1010

Write a short verification checklist for logarithm laws, then apply it to one worked example from this lesson.

Write a complete attempt before opening the exact answer.

Attempt once to unlock the answer

Complete a substantive attempt before revealing the server-held answer.

Lesson close

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.

Expandlogb(5x3y)log_b(\frac{5x^3}{y})logb​(y5x3​)

Expandlogb(5x3y)log_b(\frac{5x^3}{y})logb​(y5x3​)

Expandlogb(xy)log_b(\frac{x}{y})logb​(yx​)

Condense 3lnx−lny3lnx-lny3lnx−lny.

Original BetterGrades manuscript, rights-separated references.

Expand $log_b(\frac{5x^3}{y})$

Expand $log_b(\frac{5x^3}{y})$

Expand $log_b(\frac{x}{y})$

Condense $3lnx-lny$ .