Logarithm Properties and Equations

Explore product rule, quotient and power rules, solving and checking.

Lesson 4 of 53 parts Practice Flashcards

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Product Rule

Objective: Explain product rule and use it to solve or interpret a problem.

For positive M,N, log_b(MN)=log_b M+log_b N; multiplication inside becomes addition of logarithms, not addition inside. No, there is no general log-sum rule.

Worked examples

Try itLogarithm Properties and Equations — practice 1.1: If log_4(64)=3 and log_4(1024)=5, find log_4(64·1024).Show answer

Step 1: Use log base a of a product as the sum of logs. Step 2: Add 3+5. Step 3: The result is 8.

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Product Rule

Objective: Explain product rule and use it to solve or interpret a problem.

For positive M,N, log_b(MN)=log_b M+log_b N; multiplication inside becomes addition of logarithms, not addition inside. No, there is no general log-sum rule.

Worked examples

Try itLogarithm Properties and Equations — practice 1.1: If log_4(64)=3 and log_4(1024)=5, find log_4(64·1024).Show answer

Step 1: Use log base a of a product as the sum of logs. Step 2: Add 3+5. Step 3: The result is 8.

Quotient and Power Rules

Objective: Explain quotient and power rules and use it to solve or interpret a problem.

For positive M,N, log_b(M/N)=log_b M−log_b N and log_b(M^r)=r log_b M under valid domains. Real logarithms are undefined for zero or negative inputs.

Worked examples

Try itLogarithm Properties and Equations — practice 2.1: Evaluate log_2(1024/4).Show answer

Step 1: Use exponent quotient: 2^10/2²=2^8. Step 2: Log base 2 undoes its exponent. Step 3: The value is 8.

Solving and Checking

Objective: Explain solving and checking and use it to solve or interpret a problem.

After using exponentials to solve a log equation, verify every logarithm argument is positive in the original statement. Yes, manipulating logarithms can conceal a forbidden argument.

Worked examples

Try itLogarithm Properties and Equations — practice 3.1: Evaluate log_3((3²)^11).Show answer

Step 1: Apply exponent power rule: (3²)^11=3^22. Step 2: Log base 3 cancels that exponent. Step 3: The answer is 22.

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