Problem 9 - Algebra A Combined Function

Question

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{5} 12-\log _{5} 4 $$

Step-by-Step Solution

Answer

\( \log_5 12 - \log_5 4 = \log_5 3 \)

Step 1: Understand the Subtraction Property of Logarithms

The subtraction of two logarithms with the same base can be expressed as a single logarithm by using the property: \( \log_b A - \log_b B = \log_b \left( \frac{A}{B} \right) \). In this case, the base is 5.

Step 2: Apply the Subtraction Property

Using the property from Step 1, apply it to \( \log_5 12 - \log_5 4 \). The expression becomes \( \log_5 \left( \frac{12}{4} \right) \).

Step 3: Simplify the Fraction

Simplify the fraction inside the logarithm: \( \frac{12}{4} = 3 \). Therefore, the expression \( \log_5 \left( \frac{12}{4} \right) \) simplifies to \( \log_5 3 \).

Step 4: Write the Final Answer

The simplified form of the given expression \( \log_5 12 - \log_5 4 \) is \( \log_5 3 \).

Key Concepts

Logarithm Subtraction

Subtraction between logarithms follows a tidy rule that simplifies these expressions easily. This process is known as the subtraction property of logarithms. When you have two logarithms with the same base being subtracted, such as \(\log_b A - \log_b B\), you can combine them into a single logarithm. The property states that it equals \(\log_b \left( \frac{A}{B} \right) \).

• Same Base Requirement: Ensure that the two logarithms you're combining have the same base. This is crucial for applying the subtraction property.
• Understanding the Fraction: Essentially, if you subtract two logs, you're dividing the two quantities inside the logs.

This property is especially useful when you want to simplify an expression or solve equations involving logs.

Single Logarithm

Expressing differences of logarithms as a single logarithm is not only neat but also simplifies calculations. By changing an expression like \( \log_5 12 - \log_5 4 \) into \( \log_5 3 \), you eliminate complexities.

• Simple Combination: Transforming two logs into one removes the need to handle multiple separate terms.
• Equivalent Representation: Though the form changes, the values represented by both expressions before and after combining remain equivalent.
• Simplifies Analysis: Once in a single form, it is easier to evaluate or compare these expressions.

Thus, it's always beneficial to condense logarithmic expressions to their simplest forms.

Logarithmic Expressions

Logarithmic expressions are mathematical statements using logs to express a quantity. They often involve operations such as multiplication, division, addition, and subtraction.

• Base Understanding: In a logarithmic expression, the base determines the number the logarithm is comparing other numbers to. For example, in \( \log_5 12 \), 5 is the base.
• Variable Representation: Variables in these expressions should represent positive values since negative numbers and zero aren't within the domain of real-number logarithms.
• Practical Use: Logs are incredibly important in various fields like science and engineering because they help solve equations involving exponential growth, decay, or any scale of measure that requires orders of magnitude.

Grasping how to manipulate and simplify these expressions is key in understanding larger concepts involving logarithms and their applications.