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Chapter 9: Problem 70

Simplify. $$\log _{p} p^{-5}$$

Short Answer

Expert verified
-5

Step by step solution

01

Understand the logarithm properties

Recall that logarithms have several properties. One important property is that \(\text{log}_a(a^b) = b\).
02

Apply the property to the given problem

In the given expression \( \text{log}_p (p^{-5}) \), identify \(a=p\) and \(b=-5\). According to the property, \( \text{log}_a(a^b) = b\), we can write: \(\text{log}_p (p^{-5}) = -5\).

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Key Concepts

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

logarithms
Logarithms might seem difficult at first, but they become manageable when you understand their properties. Logarithms essentially express the idea of exponents in a different way. The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, in the equation \( \text{log}_a(b) = c \), the base \(a\) raised to the power \(c\) equals \(b\). This relationship can be written as \(a^c = b\). When working with logarithms, several properties help simplify expressions:
  • \( \text{log}_a(1) = 0\) because any number raised to the power of zero is one.
  • \( \text{log}_a(a) = 1\) because any number raised to the power of one is itself.
  • \( \text{log}_a(a^b) = b\) which shows that the logarithm of a base raised to an exponent is equal to the exponent.
These properties become very useful when simplifying complex logarithmic expressions.
exponents
Exponents represent repeated multiplication of a base number. They are expressed as \(a^b\), where \(a\) is the base and \(b\) is the exponent (or power). For example, \(2^3 = 2 \times 2 \times 2 = 8\). Understanding the rules and properties of exponents is crucial when working with logarithms:
  • Product Rule: \(a^m \times a^n = a^{m+n}\)
  • Quotient Rule: \( \frac{a^m}{a^n} = a^{m-n}\)
  • Power Rule: \( (a^m)^n = a^{mn}\)
  • Negative Exponent: \(a^{-n} = \frac{1}{a^n}\)
In the expression \( \text{log}_p (p^{-5}) \), the base \(a\) is replaced with \(p\) and the exponent \(-5\). So, when we simplify it using the logarithm property \( \text{log}_a(a^b) = b\), we get \(\text{log}_p (p^{-5}) = -5\).
simplification
Simplification is the process of reducing complex expressions into simpler forms to make them easier to work with. In the example problem, we simplified the logarithmic expression using basic properties of logarithms and exponents. Let's break this down further:
  • Identify the components of the expression. Here, \(a\) is \(p\) and \(-5\) is the exponent.
  • Apply the logarithmic property \( \text{log}_a(a^b) = b\). Using this property, \( \text{log}_p (p^{-5}) \) simplifies directly to \(-5\).
Practicing these simplifications with different expressions can help you become more familiar and proficient in mathematics. Remember to always check your work, and use these properties as tools to make complicated problems more manageable.

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