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Chapter 12: Problem 44

Find the value of each logarithmic expression. $$ \log _{3} \frac{1}{9} $$

Short Answer

Expert verified
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Step by step solution

01

Understanding Logarithmic Expression

The logarithmic expression \( \log_{3} \frac{1}{9} \) asks for the power to which the base 3 must be raised to yield \( \frac{1}{9} \).
02

Recognize the Reciprocal

Notice that \( \frac{1}{9} = \frac{1}{3^2} \). This means we need to find what exponent makes \( 3^{-2} \) out of the base 3.
03

Solve the Logarithmic Equation

Since \( \log_{3} 3^{-2} = -2 \) because raising 3 to the power of \(-2\) gives \( \frac{1}{9} \), the value of the logarithmic expression is \(-2\).
04

Verify the Result

To verify, calculate \( 3^{-2} \) which indeed is \( \frac{1}{9} \). Thus, \( \log_{3} \frac{1}{9} = -2 \) is correct.

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Key Concepts

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

Understanding Logarithms
Logarithms can seem a bit tricky at first, but they are simply another way of looking at exponents. When we see an expression like \( \log_{3} \frac{1}{9} \), the goal is to find the exponent that the base, in this case, 3, must be raised to in order to obtain \( \frac{1}{9} \).
In general, the logarithm \( \log_{b} x \) answers the question: "To what power must the base \( b \) be raised, to produce the number \( x \)?"
To make sense of logarithms:
  • Identify the base of the logarithm.
  • Determine the number you are trying to achieve with that base raised to a power.
  • Find the specific power or exponent that accomplishes this.
By unraveling these parts, you can demystify logarithmic expressions.
Exploring Exponents
Exponents are a fundamental concept in mathematics that express how many times a number, known as the base, is multiplied by itself. For example, \( 3^2 \) means 3 multiplied by 3, which equals 9. In the problem \( \log_{3} \frac{1}{9} \), we are exploring what exponent allows 3 to become \( \frac{1}{9} \).
Exponents follow specific rules such as:
  • The power of zero: Any base raised to the power of zero is 1, e.g., \( b^0 = 1 \).
  • The power of one: Any base raised to the power of one is the base itself, e.g., \( b^1 = b \).
  • Negative exponents: Negative exponents represent reciprocals. For instance, \( b^{-n} \) is equal to \( \frac{1}{b^n} \).
Understanding these rules helps in solving logarithmic questions, as in this exercise where \( 3^{-2} \) signifies \( \frac{1}{9} \).
Base and Power Relationships
Grasping the relationship between base and power is essential in both exponents and logarithms. In the logarithmic expression \( \log_{3} \frac{1}{9} \), the base is 3. We are determining what power of 3 equals \( \frac{1}{9} \), which turns out to be \(-2\).
Here's how you can think about bases and powers:
  • Identify the base: The number being raised to a power. In our exercise, this is the number 3.
  • Find the power needed: The exponent that relates the base to the outcome (here it's \(-2\) for \( \frac{1}{9} \)).
  • Relate to the result: Recognize how changing the power affects the outcome.
This base-power relationship is crucial in various math problems, allowing you to switch between exponentials and logarithms confidently.

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Most popular questions from this chapter

Find the value of each logarithmic expression. $$ \log _{2} 16 $$

Explain why 1 is not included as a logarithmic base.

If \(x=-2, y=0\), and \(z=3\), find the value of each expression. $$ \frac{x^{3}-2 y+z}{2 z} $$

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Graph each logarithmic function. $$ y=\log _{2} x $$
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