Problem 30 - Algebra A Combined Function

Question

Write each as a single logarithm. Assume that variables represent positive numbers. $$ \log _{6} 21+\log _{6} 2-\log _{6} 7 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \log_{6} 6 \).

Step 1: Apply Logarithmic Addition Rule

First, use the property of logarithms that states the sum of two logs with the same base can be combined into a single log by multiplying their arguments. Given the expression \( \log_{6} 21 + \log_{6} 2 \), apply this property: \[\log_{6} (21 \times 2) = \log_{6} 42\]

Step 2: Apply Logarithmic Subtraction Rule

Next, utilize the property that the difference of two logs with the same base can be combined into a single log by dividing their arguments. We have the expression: \[\log_{6} 42 - \log_{6} 7\]Apply the rule to get:\[\log_{6} \left( \frac{42}{7} \right)\]

Step 3: Simplify the Fraction Inside the Logarithm

Finally, simplify the fraction within the expression:\[\frac{42}{7} = 6\]Substitute back into the expression to get:\[\log_{6} 6\]

Key Concepts

Logarithmic Addition Rule

The logarithmic addition rule is an essential concept in simplifying logarithmic expressions. It states that when you have two or more logarithms with the same base added together, you can combine them into a single logarithm by multiplying their arguments. This property is mathematically represented as:

• \( \log_b x + \log_b y = \log_b (x \times y) \)

In our original exercise problem, this rule is applied to the expression \( \log_6 21 + \log_6 2 \). By using the addition rule, these two logarithms can be combined to form:

• \( \log_6 (21 \times 2) = \log_6 42 \)

This single expression, \( \log_6 42 \), represents both logarithms added together. This helps simplify complex logarithmic expressions into more manageable forms.

Logarithmic Subtraction Rule

The logarithmic subtraction rule is another vital tool in dealing with logarithmic expressions. Similar to the addition rule, this rule applies to logarithms with the same base. However, instead of multiplying, when you subtract two logarithms, you divide their arguments. The rule is defined as:

• \( \log_b x - \log_b y = \log_b \left( \frac{x}{y} \right) \)

In the exercise, we utilize this rule on the expression resulting from the addition rule application, which was \( \log_6 42 - \log_6 7 \). By applying the subtraction rule, these two logarithms are combined into:

• \( \log_6 \left( \frac{42}{7} \right) \)

This equation demonstrates how subtraction transforms into a division operation inside the logarithm, further simplifying the expression.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves the application of both the addition and subtraction rules to rewrite an expression as a single logarithm, often making it easier to evaluate or understand.Here is a straightforward process:

• Identify and apply the logarithmic addition rule to combine logarithms added to each other.
• Use the logarithmic subtraction rule for any subtraction among logarithms.
• Simplify any resulting numerical calculations inside the single logarithm.

In the final step of our exercise, after applying both rules to combine the terms, \( \log_6 \left( \frac{42}{7} \right) \) simplifies to \( \log_6 6 \), because \( \frac{42}{7} \) simplifies directly to 6. This simplification process results in the most concise representation of the original expression. This underscores the power of logarithmic rules in reducing expressions to simpler forms, which is particularly useful in mathematical problem-solving.