Convert a Product into a Sum of Logarithms

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: For positive x,y, rewrite log_3(xy). Candidate answers: (A) (log_3x)(log_3y) (B) log_3(x+y) (C) log_3x+log_3y (D) log_3x−log_3y.
• Independent check inputs: For positive x,y, rewrite log_4(xy). Candidate answers: (A) (log_4x)(log_4y) (B) log_4(x+y) (C) log_4x+log_4y (D) log_4x−log_4y.

Learn the idea

Logarithms reverse exponentiation with a valid positive base other than one. They answer exponent questions, reorganize multiplicative structure and solve exponential relationships while requiring positive real logarithm arguments. The logarithm product rule reflects the addition of exponents when same-base powers multiply. For positive factors, a product inside the logarithm becomes a sum of separate logarithms.

Here is the method on the supplied teaching case. The product rule turns multiplication inside a logarithm into addition outside. It applies with positive factors and valid base 3; it does not split a sum or multiply logarithm values.

The rule does not split a sum inside a logarithm. State positive-factor restrictions so every displayed real logarithm exists; multiplying logarithm values is not the product rule.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

Your answer is temporary and is not saved to an account. Download your current answer before leaving or reloading. Written work on paper or in another app stays in your own file.

Take it further

Changed-input task: For positive x,y, rewrite log_10(xy). Candidate answers: (A) (log_10x)(log_10y) (B) log_10(x+y) (C) log_10x+log_10y (D) log_10x−log_10y. Show the mathematical reason, not only the selected answer. Expected result: log_10x+log_10y. The product rule turns multiplication inside a logarithm into addition outside. It applies with positive factors and valid base 10; it does not split a sum or multiply logarithm values.

Nothing here is saved to an account or sent to a teacher. The answer download includes your current choice and any checked feedback. It does not include choices from the separate More practice section. Download that section separately.