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Chapter 5: Problem 14

Express as a product. $$\log _{b} Q^{-8}$$

Short Answer

Expert verified
\( -8 \log_{b} Q \)

Step by step solution

01

Understand the Problem

Given a logarithmic expression \( \log_{b} Q^{-8} \), it needs to be rewritten as a product.
02

Use Logarithm Property

Apply the logarithm power rule, which states that \( \log_{b} (x^y) = y \log_{b} x \).
03

Apply the Rule

Using the logarithm power rule, rewrite \( \log _{b} Q^{-8} \) as \( -8 \log_{b} Q \).
04

Write the Final Answer

The expression \( \log _{b} Q^{-8} \) as a product is \( -8 \log_{b} Q \).

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Key Concepts

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

Logarithmic Properties
Logarithms possess several essential properties that simplify complex expressions. Knowing these properties is crucial for solving logarithmic problems effectively.
The fundamental properties include:
  • Product Property: \( \log_{b} (xy) = \log_{b} x + \log_{b} y\)
  • Quotient Property: \( \log_{b} \left( \frac{x}{y} \right) = \log_{b} x - \log_{b} y\)
  • Power Property: \( \log_{b}(x^y) = y \log_{b} x\)
By mastering these properties, you can easily manipulate and simplify logarithmic expressions, leading to quicker and more accurate solutions. In our problem, we specifically use the power property to transform the given logarithmic expression.
Power Rule
The power rule for logarithms is relatively straightforward and immensely useful. Formally, it states: \( \log_{b} (x^y) = y \log_{b} x \)
This rule tells us that when a logarithmic argument is raised to an exponent, we can 'bring down' the exponent as a coefficient in front of the logarithm.
Let's break it down step-by-step using our exercise:
  • We start with \( \log_{b} Q^{-8} \)
  • According to the power rule, we can bring the exponent -8 to the front, making the expression \( -8 \log_{b} Q \)
This simplification makes it easier to comprehend and work with the logarithmic expression.
Expressions Rewriting
Rewriting expressions is a crucial technique in mathematics that allows us to simplify, compare, and ultimately solve problems more efficiently. By using known rules and properties like those of logarithms, we can transform complex expressions into more workable forms.
In our exercise, the initial expression was \(\log_{b} Q^{-8}\). With the help of the power rule, we rewrote it as \( -8 \log_{b} Q \).
Here is why rewriting matters:
  • It simplifies calculations by reducing complicated forms into simpler ones
  • It helps in identifying relationships and patterns within the problem
  • It makes it easier to apply further mathematical principles or solve equations
In our context, rewriting the logarithmic expression into a product form makes it manageable and paves the way for solving more complex problems involving logarithms.

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Most popular questions from this chapter

Determine whether each of the following is true or false. Assume that \(a, x, M,\) and \(N\) are positive. $$\frac{\log _{a} M}{\log _{a} N}=\log _{a} M-\log _{a} N$$

Solve using any method. $$\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=3$$

Suppose that \(\log _{a} x=2 .\) Find each of the following. $$\log _{1 / a} x$$

Solve. $$\log _{6} x=1-\log _{6}(x-5)$$

Express as a single logarithm and, if possible, simplify. $$\log _{a}\left(a^{10}-b^{10}\right)-\log _{a}(a+b)$$
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