Problem 65
Question
Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{2}{5}$$
Step-by-Step Solution
Verified Answer
\( \log_{3} 2 - \log_{3} 5 \)
1Step 1: Identify the logarithmic property
The given expression is \( \log_{3} \frac{2}{5} \). We can use the property of logarithms that states \( \log_{b} \left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N \). This allows us to separate the logarithm of a quotient into the difference of two logarithms.
2Step 2: Apply the Quotient Property
Using the property identified in Step 1, rewrite the expression: \( \log_{3} \frac{2}{5} = \log_{3} 2 - \log_{3} 5 \).
3Step 3: Write the final expression
The expression \( \log_{3} \frac{2}{5} \) has been rewritten as \( \log_{3} 2 - \log_{3} 5 \) using the properties of logarithms.
Key Concepts
Properties of LogarithmsQuotient PropertyLogarithmic Expressions
Properties of Logarithms
In mathematics, the properties of logarithms are fundamental tools for simplifying and manipulating logarithmic expressions. These properties allow us to express and solve complicated logarithmic equations with more ease. Here are some key properties:
Each property serves a unique purpose. The Product Property helps in breaking down products into sums. The Power Property allows us to pull down exponents as coefficients. These manipulations simplify complex logarithmic expressions, making them easier to work with. By understanding and applying these properties, one can effectively solve logarithmic problems.
- Product Property: \( \log_{b}(MN) = \log_{b} M + \log_{b} N \)
- Quotient Property: \( \log_{b}\left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N \)
- Power Property: \( \log_{b}(M^n) = n\log_{b} M \)
Each property serves a unique purpose. The Product Property helps in breaking down products into sums. The Power Property allows us to pull down exponents as coefficients. These manipulations simplify complex logarithmic expressions, making them easier to work with. By understanding and applying these properties, one can effectively solve logarithmic problems.
Quotient Property
The Quotient Property of logarithms is incredibly useful in simplifying expressions involving the division of two numbers inside a logarithmic function. According to this property:
The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. This can be mathematically represented as:
Imagine you have the expression \( \log_{3} \frac{2}{5} \). The Quotient Property allows you to break this into two separate logarithms: \( \log_{3} 2 - \log_{3} 5 \).
This makes it easier when you need to perform calculations or apply further mathematical operations. The property is particularly useful when dealing with complex fractions as it simplifies and separates the two involved quantities.
The logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. This can be mathematically represented as:
- \( \log_{b}\left(\frac{M}{N}\right) = \log_{b} M - \log_{b} N \)
Imagine you have the expression \( \log_{3} \frac{2}{5} \). The Quotient Property allows you to break this into two separate logarithms: \( \log_{3} 2 - \log_{3} 5 \).
This makes it easier when you need to perform calculations or apply further mathematical operations. The property is particularly useful when dealing with complex fractions as it simplifies and separates the two involved quantities.
Logarithmic Expressions
Logarithmic expressions involve the use of logarithms in mathematical equations or functions. These can arise in various fields such as science, engineering, and economics due to their relationship with exponential growth and decay.
To work with logarithmic expressions, one must be comfortable with the underlying properties of logarithms, as these provide the tools necessary for manipulation and simplification.
Understanding how to rewrite and simplify logarithmic expressions is crucial for solving equations in which logarithms appear. This understanding helps prevent errors and ensures faster problem-solving. The ability to transform and manipulate these expressions efficiently is a powerful skill in mathematics.
To work with logarithmic expressions, one must be comfortable with the underlying properties of logarithms, as these provide the tools necessary for manipulation and simplification.
- A typical expression might look like: \( \log_{b}x \)
- Expressions can often be simplified using logarithmic properties, as seen with \( \log_{3} \frac{2}{5} = \log_{3} 2 - \log_{3} 5 \)
Understanding how to rewrite and simplify logarithmic expressions is crucial for solving equations in which logarithms appear. This understanding helps prevent errors and ensures faster problem-solving. The ability to transform and manipulate these expressions efficiently is a powerful skill in mathematics.
Other exercises in this chapter
Problem 65
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