Problem 23 - Algebra 2 and Trigonometry

Question

a. Expand each expression as a difference, sum, and/or multiple of logarithms. b. Find the value of each expression. \(\log _{4} \sqrt{16^{2}}\)

Step-by-Step Solution

Answer

The expanded form is \( 2 \cdot \log_4(4) \), and the value of the expression is 2.

Step 1: Simplify the Expression Inside the Logarithm

Start by simplifying the expression inside the logarithm: \( \sqrt{16^2} \). Since \( \sqrt{16^2} = 16 \), thus the expression becomes \( \log_4 16 \).

Step 2: Express the Number as a Power of the Log Base

Recognize that \( 16 \) can be rewritten as a power of \( 4 \). Since \( 16 = 4^2 \), the expression is now \( \log_4(4^2) \).

Step 3: Apply the Power Rule for Logarithms

Use the power rule for logarithms, \( \log_b(a^n) = n \cdot \log_b(a) \), on the expression. Thus, \( \log_4(4^2) = 2 \cdot \log_4(4) \).

Step 4: Evaluate the Logarithm

Recognize that \( \log_4(4) = 1 \) since the base and the argument are the same: \( 4^1 = 4 \). Therefore, \( 2 \cdot \log_4(4) = 2 \cdot 1 = 2 \).

Key Concepts

Power Rule for Logarithms

Logarithms have a set of rules that govern how they can be computed and manipulated. One of these is the power rule for logarithms. This rule states that if you have a logarithm of a number that is raised to a power, you can bring the exponent down in front of the logarithm as a multiplier. Formally, this is represented as \( \log_b(a^n) = n \cdot \log_b(a) \).

In simpler terms, the power becomes a coefficient of the logarithm. This can make complex logarithmic expressions much easier to simplify. For example, consider \( \log_4(4^2) \). According to the power rule, the expression simplifies to \( 2 \cdot \log_4(4) \).

Once we've simplified using the power rule, it's easier to further evaluate the logarithm. Remember that applying the power rule can also make calculations more efficient by reducing large numbers into manageable forms.

Logarithmic Expressions

Logarithmic expressions involve any expression where a logarithm is a main operation. Understanding these expressions starts with recognizing the base and the argument of the logarithm. For a logarithm \( \log_b(a) \), \( b \) is the base and \( a \) is the argument.

When dealing with these expressions, it is essential to check if the argument can be re-written in a form related to the base. For instance, in \( \log_4(16) \), 16 can be written as \( 4^2 \).

Rewriting arguments as powers of the base simplifies the expression significantly and often allows the use of the power rule effectively. This simplification follows a logical structure that makes further computation or evaluation more straightforward. Always ensure you simplify the inside of the logarithmic expression as much as possible before proceeding to evaluate.

Simplifying Expressions

Simplifying expressions is a critical skill in mathematics, especially when dealing with logarithms. Simplification involves reducing an expression to its most basic form while maintaining the same value.

In logarithmic expressions, this may mean:

• Rationalizing complex expressions (e.g., simplifying \( \sqrt{16^2} \) to 16).
• Expressing numbers as powers of the base to apply power rules.
• Combining or separating logarithms through rules of logarithms, such as the product, quotient, and power rules.

The main goal is to find a form that is more convenient for calculation. Simplification often leads to final expressions that are easier to evaluate, such as finding \( 2 \cdot \log_4(4) = 2 \) in the described exercise.

By consistently simplifying expressions, you make complex mathematical tasks more manageable and less prone to errors. Simplification is essentially about seeing the underlying structure of expressions and transforming them to reveal their simplest forms.