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Chapter 9: Problem 23

Express as an equivalent expression that is a difference of two logarithms. $$\log _{2} \frac{5}{11}$$

Short Answer

Expert verified
\( \log _{2} \frac{5}{11} = \log _{2} 5 - \log _{2} 11 \)

Step by step solution

01

Understand the Logarithm Property

To express \( \log _{2} \frac{5}{11} \) as a difference of two logarithms, use the property of logarithms that states \( \log_b \frac{M}{N} = \log_b M - \log_b N \).
02

Apply the Property

Apply the property to separate the logarithms of the numerator and the denominator. \( \log _{2} \frac{5}{11} = \log _{2} 5 - \log _{2} 11 \).
03

Simplify

The equivalent expression for \( \log _{2} \frac{5}{11} \) is \( \log _{2} 5 - \log _{2} 11 \). This is the simplified form expressed as a difference of two logarithms.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

difference of logarithms
The concept of the difference of logarithms is based on the logarithmic property that allows us to express the logarithm of a quotient as the difference of two logarithms. Specifically, the rule states: \( \text{log}_b \frac{M}{N} = \text{log}_b M - \text{log}_b N \)This means that when you have the logarithm of a fraction such as \( \text{log}_b \frac{M}{N} \), you can break it down into the difference between the logarithm of the numerator (M) and the logarithm of the denominator (N).
logarithm rules
Logarithm rules, also known as logarithmic properties, are essential tools in algebra and higher mathematics. Here are the primary rules you need to know:
  • Product Rule: \( \text{log}_b (MN) = \text{log}_b M + \text{log}_b N \)
  • Quotient Rule: \( \text{log}_b \frac{M}{N} = \text{log}_b M - \text{log}_b N \)
  • Power Rule: \( \text{log}_b (M^k) = k \text{log}_b M \)
These rules help simplify complex logarithmic expressions and solve logarithmic equations. They are derived from the properties of exponents and apply to logarithms of any base, provided the base is positive and not equal to one. To master logarithms, ensure you're comfortable with these rules and how to apply them in various scenarios.
intermediate algebra
Understanding logarithm properties is crucial in intermediate algebra. Logarithms are the inverses of exponential functions, which means they can be used to solve equations where the variable is an exponent. Mastering logarithmic properties helps students tackle different algebraic problems, including simplifying expressions and solving equations.In intermediate algebra, you'll encounter problems that require you to:
  • Simplify logarithmic expressions using the quotient, product, and power rules.
  • Solve logarithmic equations by isolating the logarithmic term and converting to exponential form.
  • Understand and apply properties of logarithms to real-world problems, such as calculating compound interest and population growth.
These skills are foundational for more advanced mathematics and various applications in science, engineering, and finance. Practice and familiarity with these concepts will make higher-level math more approachable and less intimidating.

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