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Chapter 12: Problem 68

Simplify. $$\log _{Q} Q^{-2}$$

Short Answer

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Step by step solution

01

Recognize the properties of logarithms

Use the property of logarithms that states \(\log_b(b^x) = x\). This means if the base of the logarithm and the base of the exponent are the same, the logarithm simplifies to the exponent.
02

Apply the property to the given expression

Identify that the base of the logarithm is \(Q\) and the argument is \(Q^{-2}\). Therefore, apply the property: \(\log_Q(Q^{-2}) = -2\)

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Key Concepts

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

logarithm properties
Understanding the properties of logarithms is essential for simplifying logarithmic expressions. One of the most useful properties is the identity \(\text{log}_b(b^x) = x\). This tells us that the logarithm of a number with a matching base exponent is simply the exponent itself.
  • Base: The number that is raised to a power.
  • Exponent: The power to which the base is raised.
For example, \(\text{log}_2(2^3) = 3\), because the base (2) with exponent 3 simplifies the logarithm to 3. This property can make otherwise complex problems straightforward.
Always make sure that the base of the logarithm matches the base of the exponent in the expression you are simplifying.
exponents
Exponents are fundamental in mathematics and are closely related to logarithms. An exponent indicates how many times a number, known as the base, is multiplied by itself. For instance, \2^3 = 2 \times 2 \times 2 = 8\.
Some important points about exponents include:
  • Multiplying powers with the same base: \(a^m \times a^n = a^{m+n}\)
  • Dividing powers with the same base: \(a^m / a^n = a^{m-n}\)
  • Raising a power to another power: \((a^m)^n = a^{mn}\)
      • In the given exercise, the specific exponent we deal with is -2.
      This transforms the base Q into an expression: \(Q^{-2} = \frac{1}{Q^2}\). The negative exponent means our base Q is being 'flipped' to the reciprocal and then squared.
logarithmic functions
Logarithms are essentially the reverse of exponents and are a fundamental concept in many areas of mathematics. A logarithmic function is written as \(\text{log}_b(x)\), where b is the base and x is the argument.
For example, \(\text{log}_2(8) = 3\) because \2^3 = 8\.
Some critical properties of logarithmic functions include:
  • The product property: \(\text{log}_b(xy) = \text{log}_b(x) + \text{log}_b(y)\)
  • The quotient property: \(\text{log}_b(\frac{x}{y}) = \text{log}_b(x) - \text{log}_b(y)\)
  • The power property: \(\text{log}_b(x^p) = p \times \text{log}_b(x)\)
In our exercise, the logarithmic function simplifies because the base of the logarithm and the base of the exponent match (both are Q).
Therefore, using the logarithm property, \(log_Q(Q^{-2}) = -2\). Understanding these elements makes solving logarithmic equations and simplifying expressions much easier.

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