Problem 25 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log _{2}\left(A B^{2}\right) $$

Step-by-Step Solution

Answer

\( \log_{2} A + 2 \cdot \log_{2} B \)

Step 1: Identify the Expression

We start with the expression given in the problem: \( \log _{2}(A B^{2}) \). Our task is to apply the Laws of Logarithms to expand this expression.

Step 2: Apply the Product Rule of Logarithms

According to the product rule of logarithms, \( \log_b(xy) = \log_b x + \log_b y \). We apply this rule to \( \log _{2}(A B^{2}) \), resulting in: \( \log_{2} A + \log_{2} B^2 \).

Step 3: Apply the Power Rule of Logarithms

The power rule of logarithms states that \( \log_b(x^n) = n \cdot \log_b x \). We apply this to \( \log_{2} B^2 \) which gives us: \( 2 \cdot \log_{2} B \).

Step 4: Combine the Expanded Parts

Combine the results from the previous steps: \( \log_{2} A + 2 \cdot \log_{2} B \). This is the fully expanded form of the original logarithmic expression.

Key Concepts

Product Rule of Logarithms

The Product Rule of Logarithms is a handy tool when working with logarithmic expressions involving multiplication. It states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. This can be represented as follows: if you have \( \log_b(xy) \), you can break it down into \( \log_b x + \log_b y \). In our given exercise, the expression \( \log_2(A B^2) \) uses this rule. Here, we have two components inside the logarithm, \( A \) and \( B^2 \), multiplied together. By applying the product rule, we separate these components into two distinct logarithmic expressions: \( \log_2 A + \log_2 B^2 \). This transformation helps simplify the expression and make it more tractable for further manipulation.

Power Rule of Logarithms

The Power Rule of Logarithms is crucial when dealing with exponents within a logarithmic expression. It allows you to "bring down" the exponent in front of the logarithm. Mathematically, the rule can be written as: if you have \( \log_b(x^n) \), it simplifies to \( n \cdot \log_b x \). This is especially useful for complex calculations.In our case, after using the product rule, we encounter the term \( \log_2 B^2 \). By employing the power rule, we can simplify this to \( 2 \cdot \log_2 B \). Here, the exponent "2" is brought to the front as a multiplier, making the expression simpler and easier to interpret.This rule can be applied repeatedly for different parts of an expression, allowing for comprehensive simplification and understanding of logarithmic functions.

Logarithmic Expansion

Logarithmic Expansion is the process of breaking down complex logarithmic expressions into simpler components using logarithm laws. The main goal is to make expressions more manageable and solvable. This involves systematically applying the Laws of Logarithms, such as the product, quotient, and power rules.In the exercise, expanding the expression \( \log_2(A B^2) \) starts with the Product Rule, dividing the product into addition: \( \log_2 A + \log_2 B^2 \). Next, by applying the Power Rule to \( \log_2 B^2 \), we further expand it to \( 2 \cdot \log_2 B \). After applying these rules, we achieve the final expanded form: \( \log_2 A + 2 \cdot \log_2 B \). This expanded format is easier to evaluate for various values of \( A \) and \( B \), showcasing the versatility and power of logarithmic expansion. Understanding how to methodically expand expressions is a key skill in the study of logarithms.