Problem 25 - College Algebra with Modeling and Visualization

Question

Simplify the expression. $$ \log _{1 / 3}\left(\frac{1}{3}\right)^{64} $$

Step-by-Step Solution

Answer

The expression simplifies to 64.

Step 1: Understand the Expression

The expression given is \( \log _{1 / 3}\left(\frac{1}{3}\right)^{64} \). This is a logarithmic expression where the base is \( \frac{1}{3} \) and the argument is \( \left(\frac{1}{3}\right)^{64} \).

Step 2: Apply Logarithmic Identity

Use the logarithmic identity \( \log_b(a^n) = n \log_b(a) \). For this expression, \( a = \frac{1}{3} \), \( n = 64 \), and \( b = \frac{1}{3} \). Thus, it becomes \( 64 \log_{1/3}\left(\frac{1}{3}\right) \).

Step 3: Solve the Logarithmic Function

Since \( \log_b(b) = 1 \), we have \( \log_{1/3}(1/3) = 1 \). Substitute this into the expression from Step 2 to get \( 64 \times 1 \).

Step 4: Simplify the Result

Multiply \( 64 \times 1 \) to get the simplified result, which is 64.

Key Concepts

Logarithmic Expressions

Logarithmic expressions describe a way of writing down numbers based on their exponents. At their core, these expressions relate two numbers through a base raised to a power or exponent, which results in the original number we are interested in. In college algebra, these expressions are fundamental for solving many types of equations, particularly those involving exponential growth or decay.

Understanding logarithmic expressions starts with comprehending what a logarithm is: it answers the question, "What exponent do I need to raise the base to get the argument?" For example, in the expression \( \log_b(a) \), \( b \) is the base, and \( a \) is the argument. This equation reads as "to what power must \( b \) be raised, to yield \( a \)?"

Another important aspect of logarithmic expressions is determining the base. Common bases in mathematics include base 10, base \( e \) (natural logarithms), or any other positive number. In our example, the base was \( \frac{1}{3} \), which is less common and makes the exercise a bit more challenging.

Simplification of Expressions

Simplifying mathematical expressions involves condensing them to their simplest form, making them easier to understand or solve. For logarithmic expressions, simplification often means applying properties and identities of logarithms.

In our example, simplifying \( \log_{1/3}((\frac{1}{3})^{64}) \) involves recognizing that you can break down the exponent part of the argument using the logarithmic power rule: \( \log_b(a^n) = n \cdot \log_b(a) \). This rule essentially lets us "pull down" the exponent in front of the log, turning a potentially complex expression into a simpler multiplication problem.

Once you've applied relevant identities or rules, the task of simplification often involves basic arithmetic or algebra, like multiplying numbers or reducing fractions. It minimizes the steps or complexity inherent in the original expression, yielding results that are more straightforward to interpret and use.

Logarithmic Identities

Logarithmic identities are essential tools that simplify solving logarithmic expressions. They consist of rules and properties that define the behavior and operations involving logarithms. Some key identities frequently used in college algebra include:

• Product Rule: \( \log_b(mn) = \log_b(m) + \log_b(n) \)
• Quotient Rule: \( \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \)
• Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \)
• Change of Base Formula: \( \log_b(a) = \frac{\log_k(a)}{\log_k(b)} \), useful for converting logarithms to a more convenient base.

In the original exercise, the power rule was particularly useful. It allowed for converting the base \( \frac{1}{3} \) and the exponent 64 into a simple multiplication. Another identity applied was \( \log_b(b) = 1 \), which means any logarithm that has the argument equal to the base simplifies directly to 1. Recognizing and applying these identities can turn a potentially confusing logarithmic problem into a manageable equation to solve.