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Chapter 4: Problem 3

Simplify to a single logarithm, using logarithm properties. $$ -\log _{3}\left(\frac{1}{7}\right) $$

Short Answer

Expert verified
\(\log_{3}(7)\)

Step by step solution

01

Understand the Expression

The expression to simplify is \(-\log_{3}\left(\frac{1}{7}\right)\). It involves a negative sign and a fraction inside the logarithm.
02

Apply the Negative Logarithm Property

Use the property of logarithms that states \(-\log_{b}(x) = \log_{b}\left(\frac{1}{x}\right)\). Apply this to the given expression: \[-\log_{3}\left(\frac{1}{7}\right) = \log_{3}\left(\frac{1}{\left(\frac{1}{7}\right)}\right)\] This simplifies to \(\log_{3}(7)\).
03

Simplify the Reciprocal

Simplify the expression \(\frac{1}{\left(\frac{1}{7}\right)}\) to get 7, because taking the reciprocal of a fraction \(\frac{1}{n}\) gives you \(n\).
04

Final Simplification to Single Logarithm

Finally, the expression simplifies to a single logarithm:\[\log_{3}(7)\] This is the simplified form.

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Key Concepts

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

Simplify Logarithmic Expressions
Simplifying logarithmic expressions is a fundamental skill in algebra and calculus. When simplifying, you aim to condense logarithms into their simplest form.

The key to simplification lies in using logarithmic properties effectively. Consider the expression
  • Combining multiple logarithmic terms using properties like the product rule: \( \log_b(xy) = \log_b(x) + \log_b(y) \).
  • The quotient rule: \( \log_b\left(\frac{x}{y}\right) = \log_b(x) - \log_b(y) \), which helps in dealing with divisions.
  • And the power rule: \( \log_b(x^n) = n \cdot \log_b(x) \), which deals with exponents inside a logarithm expression.
Using these properties, expressions can be rewritten in simpler terms. For instance, converting a negative logarithmic expression using the negative logarithm property can simplify it effectively.
Negative Logarithms
Negative logarithms can seem challenging, but they're simpler when you understand their properties. A negative logarithm like \( -\log_b(x) \) can be rewritten by taking the reciprocal of its argument:
  • The property states \( -\log_b(x) = \log_b\left(\frac{1}{x}\right) \).
  • This means a negative logarithm essentially inverts the input value inside the logarithm.
For example, the expression \(-\log_3\left(\frac{1}{7}\right)\) can be converted using this property to become \(\log_3\left(\frac{1}{\left(\frac{1}{7}\right)}\right)\), which simplifies to a positive logarithm: \(\log_3(7)\). Understanding this property allows for easier manipulation of logarithmic expressions involving negative signs.
Reciprocal Properties
The reciprocal properties are pivotal in simplifying expressions involving fractions, especially in logarithms. Understanding what a reciprocal is, and how it behaves, aids significantly in simplification tasks.

Consider the reciprocal: if you flip a fraction like \( \frac{1}{n} \), you get \( n \) itself.

When dealing with expressions such as \( \frac{1}{\left(\frac{1}{7}\right)} \), applying this concept, you simply get 7. This property translates into the world of logarithms as well:
  • In the context of negative logarithms, using this property ensures you transition effortlessly from \( \log_b\left(\frac{1}{x}\right) \) to \( \log_b(x) \).
  • For example, returning to the expression \(-\log_3\left(\frac{1}{7}\right) \), recognizing it as \( \log_3(7) \) follows directly from acknowledging that the reciprocal of \( \frac{1}{7} \) is 7.
Mastering reciprocal properties equips you with the tools needed to tackle not just logarithms, but a wide range of mathematical expressions efficiently.

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Most popular questions from this chapter

Simplify to a single logarithm, using logarithm properties. $$ -\log _{4}\left(\frac{1}{5}\right) $$

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