Problem 44 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

-2

Step 1: 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 \( \frac{1}{9} \).

Step 2: 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.

Step 3: 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 \(-2\).

Step 4: Verify the Result

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

Key Concepts

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.