Logarithm Properties and Equations

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

Lesson 3 of 43 parts Practice Flashcards

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Product Rule

Objective: Explain product rule, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$log_b(MN)=log_b M+log_b N, M,N>0$

Worked examples

Try itIf log_3(243)=5 and log_3(2187)=7, find log_3(243·2187).Show answer

Step 1: Use log base a of a product as the sum of logs. Step 2: Add 5+7. Step 3: The result is 12.

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Product Rule

Objective: Explain product rule, apply the rule, and justify relevant restrictions.

Formulas

$log_b(MN)=log_b M+log_b N, M,N>0$

Worked examples

Try itIf log_3(243)=5 and log_3(2187)=7, find log_3(243·2187).Show answer

Step 1: Use log base a of a product as the sum of logs. Step 2: Add 5+7. Step 3: The result is 12.

Quotient and Power Rules

Objective: Explain quotient and power rules, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$log_b(M/N)=log_b M−log_b N, M,N>0$

Worked examples

Try itEvaluate log_3(19683/9).Show answer

Step 1: Use exponent quotient: 3^9/3²=3^7. Step 2: Log base 3 undoes its exponent. Step 3: The value is 7.

Solving and Checking

Objective: Explain solving and checking, apply the rule, and justify relevant restrictions.

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. Identify the stated domain before using the rule, preserve restrictions during algebraic simplification, and interpret the result in the units or graph features requested. Use the worked example to connect the symbolic rule with a concrete calculation.

Formulas

$log_b u=v ⇔ u=b^v$

Worked examples

Try itSolve log_3(x−5)=7.Show answer

Step 1: Rewrite as x−5=3^7. Step 2: Add 5: x=2192. Step 3: Check the argument: x−5=2187>0.

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