Problem 52 - Algebra A Combined Function

Question

Solve. $$ \log _{3} \frac{1}{81}=x $$

Step-by-Step Solution

Answer

\( x = -4 \)

Step 1: Rewrite the Equation Using Properties of Logarithms

Since the base of the logarithm is 3, we can use the property that \( \log_b \left( \frac{1}{a} \right) = -\log_b(a) \). Applying this property, the equation becomes \( \log_3 \left( \frac{1}{81} \right) = -\log_3(81) \).

Step 2: Express 81 in Terms of the Base

Notice that 81 is a power of 3. Specifically, \( 81 = 3^4 \). Substitute 81 with \( 3^4 \) in the equation from Step 1 to yield \( -\log_3(3^4) \).

Step 3: Apply the Power Rule

Use the power rule of logarithms, which states that \( \log_b(a^n) = n \cdot \log_b(a) \). Applying it here, we have \( -\log_3(3^4) = -4 \cdot \log_3(3) \).

Step 4: Simplify the Equation

Since \( \log_3(3) = 1 \), the equation simplifies to \( -4 \cdot 1 = -4 \). Thus, \( x = -4 \).

Step 5: Conclude the Solution

The value of \( x \) that satisfies the equation \( \log _{3} \frac{1}{81}=x \) is \( -4 \).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential in simplifying logarithmic expressions and solving logarithmic equations. There are several key properties that you should know:

• Product Property: This property states that the log of a product is the sum of the logs. Mathematically, \( \log_b(mn) = \log_b(m) + \log_b(n) \).
• Quotient Property: The log of a quotient is the difference of the logs. It's given by \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \).
• Power Property: This states that the log of an exponentiated number is the exponent times the log of the base number. It's expressed as \( \log_b(a^n) = n \cdot \log_b(a) \).

In the exercise, the use of the property \( \log_b \left( \frac{1}{a} \right) = -\log_b(a) \) helped transform the expression \( \log_3 \left( \frac{1}{81} \right) \). Properties like these simplify complex expressions, making it easier to find solutions.

Exponentiation

Exponentiation is a mathematical operation that is essential to understand when working with logarithms. It involves raising a number, called the base, to a power, which is the exponent. Learn more about this concept:

• Definition: The expression \( b^n \) means that the base \( b \) is multiplied by itself \( n \) times.
• Example: \( 3^4 = 3 \times 3 \times 3 \times 3 = 81 \)
• Inverse Operation: Exponentiation and logarithms are inverse operations. If \( b^x = a \), then \( \log_b(a) = x \).

In the step-by-step solution of the exercise, understanding exponentiation was crucial because it allowed us to express the number 81 as \( 3^4 \). Recognizing numbers as powers of the base in logarithmic expressions is a valuable skill for simplifying and solving equations.

Simplifying Logarithms

Simplifying logarithms involves using properties and operations to transform a complex logarithmic expression into a simpler or more useful form. Here are some steps to help you simplify logarithms effectively:

• Identify Familiar Bases and Exponents: Look for numbers that can be rewritten as powers of a common base.
• Apply Logarithm Properties: Use the properties such as product, quotient, and power to break down cumbersome expressions.
• Combine and Simplify: Combine like terms and simplify so that the expression is reduced to its simplest form.

In our problem, simplification was done by recognizing that 81 is \( 3^4 \), and then using the power rule, \( \log_3(3^4) = 4 \log_3(3) \). Further simplification was possible since \( \log_3(3) = 1 \), leading to the final answer of \( x = -4 \). Simplification techniques are key not just in solving equations, but also in making expressions more manageable and comprehensible.