Problem 40 - College Algebra with Modeling and Visualization

Question

(Refer to Examples 5 and \(6 .\) ) Write the expression as a logarithm of a single expression. $$ \log _{3} 5-\log _{3} 10-\log _{3} \frac{1}{2} $$

Step-by-Step Solution

Answer

The expression simplifies to 0.

Step 1: Apply the Quotient Rule

The quotient rule of logarithms states: \( \log_b{a} - \log_b{c} = \log_b{\frac{a}{c}} \). Apply this rule to the first two terms: \( \log_{3} 5 - \log_{3} 10 = \log_3{\frac{5}{10}} = \log_3{\frac{1}{2}} \).

Step 2: Apply the Quotient Rule Again

Now apply the quotient rule to the result from Step 1 and the third term. We have: \( \log_3{\frac{1}{2}} - \log_3{\frac{1}{2}} = \log_3{\frac{\frac{1}{2}}{\frac{1}{2}}} \).

Step 3: Simplify the Expression

The expression \( \log_3{\frac{\frac{1}{2}}{\frac{1}{2}}} \) simplifies to \( \log_3{1} \). Since the logarithm of one in any base is zero, this simplifies further to: \( 0 \).

Key Concepts

quotient rule

The quotient rule for logarithms is a very handy tool when you have subtraction in logarithmic expressions. It says that if you have \( \log_b{A} - \log_b{B} \), you can rewrite it as \( \log_b{\frac{A}{B}} \). This allows you to combine two logarithmic terms into one.

• Why is this helpful? Simplifying logarithmic expressions often involves combining terms to make the expression easier to handle or understand.
• When using the quotient rule, always ensure the logs have the same base.

In our exercise, this rule was first applied to \( \log_{3} 5 - \log_{3} 10 \), converting it into \( \log_3{\frac{5}{10}} \). This simplification transformed two separate terms into one, utilizing the beauty of the quotient rule.

logarithmic expressions

Logarithms are the inverse operations of exponents, revealing how many times a number called the base must be multiplied by itself to achieve another number. When working with logarithmic expressions, especially in subtraction, the quotient rule becomes invaluable for simplification.

• The given problem involves subtracting logarithms which signals a potential for using the quotient rule.
• A logarithmic expression like \( \log_b{a} - \log_b{c} \) simplifies into one single term \( \log_b{\frac{a}{c}} \).

Through manipulation, such expressions reveal the properties of the logarithms involved, showcasing powerful patterns and simplifications achievable.

simplifying logarithms

Simplifying logarithms involves converting them into a form that is easier to handle or interpret. In the original exercise, the goal was to express the given terms as a single logarithm. Here's how it works:- Begin with the initial expression: \( \log_{3} 5-\log_{3} 10-\log_{3} \frac{1}{2} \).- Using the quotient rule twice, this simplifies down to \( \log_3{\frac{\frac{1}{2}}{\frac{1}{2}}} \).- Further simplification yields \( \log_3{1} \).Now, recall that the logarithm of 1 in any base is 0. This is a significant aspect of simplifying logarithms: regular patterns emerge, such as knowing certain logarithmic expressions equate to zero. This simplification not only reduces complexity but makes mathematical evaluation swift and direct.