Problem 31 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{7} 6+\log _{7} 3-\log _{7} 4 $$

Step-by-Step Solution

Answer

\( \log_{7} \left( \frac{9}{2} \right) \)

Step 1: Apply the Product Rule for Logarithms

The product rule for logarithms states that \( \log_b x + \log_b y = \log_b (x \cdot y) \). Apply this rule to the first two terms: \( \log_{7} 6 + \log_{7} 3 = \log_{7} (6 \times 3) = \log_{7} 18 \).

Step 2: Apply the Quotient Rule for Logarithms

The quotient rule for logarithms states that \( \log_b x - \log_b y = \log_b \left( \frac{x}{y} \right) \). Apply this rule to the result from Step 1 and the third term: \( \log_{7} 18 - \log_{7} 4 = \log_{7} \left( \frac{18}{4} \right) \).

Step 3: Simplify the Fraction

Simplify the fraction \( \frac{18}{4} \): \( \frac{18}{4} = \frac{9}{2} \). Thus, \( \log_{7} \left( \frac{18}{4} \right) = \log_{7} \left( \frac{9}{2} \right) \).

Key Concepts

Product Rule for Logarithms

The product rule for logarithms is a handy concept when you're working with logarithmic expressions involving addition. This rule states that you can combine two logarithms into a single one if they're added, given that both have the same base. Formally, it is expressed as: \( \log_b x + \log_b y = \log_b (x \cdot y) \).
This means if you have two numbers that are in logarithmic form and added together, you can simplify them by multiplying the numbers themselves first and then taking the logarithm of the result.

• For example, consider \( \log_{7} 6 + \log_{7} 3 \). According to the product rule, this simplifies to \( \log_{7} (6 \times 3) \).
• The operation results in \( \log_{7} 18 \) because 6 times 3 equals 18.

Using this rule can greatly simplify calculations in logarithmic expressions and makes handling of equations that involve addition of logs straightforward.

Quotient Rule for Logarithms

The quotient rule for logarithms is similar to the product rule but applies to subtraction of logs. This rule tells us that the subtraction of two logarithms with the same base results in the division of their arguments. The rule is written as: \( \log_b x - \log_b y = \log_b \left( \frac{x}{y} \right) \).
In simpler terms, it means when you're subtracting logs, you can combine their arguments by dividing, forming a single logarithm.

• In our ongoing example, after using the product rule, we have \( \log_{7} 18 \), and we subtract \( \log_{7} 4 \).
• By applying the quotient rule, it simplifies to \( \log_{7} \left( \frac{18}{4} \right) \).

This approach helps in transforming a complex expression into a more manageable one, making problem-solving with logarithms efficient.

Simplifying Fractions

Simplifying fractions is a basic arithmetic skill that simplifies expressions for easier calculation and solution. Whenever you encounter fractions, look for a common factor between the numerator and the denominator to reduce the fraction to its simplest form.

• For instance, when faced with \( \frac{18}{4} \), notice that both 18 and 4 can be divided by 2.
• Dividing each by 2 results in \( \frac{9}{2} \), which is the simplest form of the fraction.

Simplifying fractions not only makes subsequent calculations easier but also helps in achieving the most straightforward version of a mathematical expression. In our logarithmic expression example, simplifying the fraction \( \frac{18}{4} \) assists in presenting the final result as \( \log_{7} \left( \frac{9}{2} \right) \), a more elegant and simplified logarithmic expression.