Problem 12 - Algebra A Combined Function

Question

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{5} 12-\log _{5} 3 $$

Step-by-Step Solution

Answer

\( \log_5 4 \).

Step 1: Identify the Logarithm Rule

To combine logarithms into a single logarithm, we use the property of logarithms which relates subtraction to division: \( \log_b A - \log_b B = \log_b \left( \frac{A}{B} \right) \).

Step 2: Apply the Rule

Apply the logarithm subtraction rule to the given expression: \[ \log_5 12 - \log_5 3 = \log_5 \left( \frac{12}{3} \right) \].

Step 3: Simplify the Expression

Simplify the fraction inside the logarithm: \( \frac{12}{3} \) simplifies to 4. Substitute this back into the logarithm: \( \log_5 \left( 4 \right) \).

Step 4: Write the Final Answer

The simplified expression as a single logarithm is \( \log_5 4 \).

Key Concepts

Logarithmic Subtraction

Logarithmic subtraction is a straightforward concept that connects logarithms with division. When you see the subtraction of two logarithms having the same base, you can use the property: - If you have an expression like \( \log_b A - \log_b B \), it can be rewritten as \( \log_b \left( \frac{A}{B} \right) \). This rule essentially tells us that when subtracting two logs of the same base, you can "condense" the logs into a single logarithm by dividing the arguments of the logarithms. For example, in the exercise \( \log_5 12 - \log_5 3 \), consider:

• Both logs are base 5, allowing the use of the subtraction rule.
• By subtracting the logs, you create a fraction within a single log: \( \log_5 \left( \frac{12}{3} \right) \).

This simplification process helps in solving logarithmic equations and is applicable in various mathematical contexts.

Combining Logarithms

Combining logarithms involves manipulating the logs into a single expression, based on their operations. Subtraction indicates that you will combine the terms through division, following the law: - \( \log_b A - \log_b B = \log_b \left( \frac{A}{B} \right) \). This concept simplifies calculations and is very useful because it reduces complexity. In the step-by-step solution:

• The original problem, \( \log_5 12 - \log_5 3 \), is handled by treating 12 and 3 as components of a division.
• Combining the logs gives: \( \log_5 \left( \frac{12}{3} \right) \).

Combining logs this way is powerful. It condenses longer logarithmic expressions, making them easier to evaluate or simplify.

Logarithm Simplification

Simplifying logarithms often involves calculating the result of operations inside the log expression. Once you have combined the logarithms, you may need to simplify the fraction or expression within the log: - Take \( \log_5 \left( \frac{12}{3} \right) \), which simplifies to \( \log_5 4 \) because \( \frac{12}{3} = 4 \). Simplifying makes the expression more digestible and sets it up for easy evaluation if needed.

• When you simplify, you're directly reducing complex terms into more straightforward, comparable identities.
• It helps in directly applying logarithmic tables or calculators if further evaluation is necessary.

This step is crucial as it finalizes the conversion from multiple logarithms into a streamlined single logarithmic expression, ready for interpretation or calculation.