Problem 19 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

\( \log_{2}(AB^2) = \log_{2}(A) + 2 \log_{2}(B) \).

Step 1: Identifying the Expression

The expression given is \( \log_{2}(AB^2) \). This is a logarithm of a product, which can be expanded using logarithm properties.

Step 2: Applying the Product Rule

According to the Product Rule of Logarithms, \( \log_b(xy) = \log_b(x) + \log_b(y) \). Apply this rule to the expression: \( \log_{2}(AB^2) = \log_{2}(A) + \log_{2}(B^2) \).

Step 3: Applying the Power Rule

The Power Rule of Logarithms states that \( \log_b(x^n) = n \log_b(x) \). Apply this to \( \log_{2}(B^2) \): \( \log_{2}(B^2) = 2 \log_{2}(B) \).

Step 4: Final Expansion

Substitute the result from the Power Rule back into the expression: \( \log_{2}(AB^2) = \log_{2}(A) + 2 \log_{2}(B) \).

Key Concepts

Product Rule of Logarithms

The Product Rule of Logarithms is an important tool when dealing with logarithms of products. It helps simplify expressions by breaking them down. The rule states that for any positive numbers \( x \) and \( y \):

• \( \log_b(xy) = \log_b(x) + \log_b(y) \)

This means we can replace the logarithm of a product with a sum of two separate logarithms. Think of it like spreading out the factors of a multiplication.
In our exercise, we expanded \( \log_2(AB^2) \) by using the product rule. We treated \( A \) and \( B^2 \) as separate entities. This way, the expression \( \log_2(AB^2) \) becomes \( \log_2(A) + \log_2(B^2) \).
This simplification makes complex logarithmic equations easier to handle and solve.

Power Rule of Logarithms

The Power Rule of Logarithms allows you to take the exponent of a variable out in front of the logarithm. This is useful when you have a variable raised to a power within a logarithm. The Power Rule is expressed as:

• \( \log_b(x^n) = n \log_b(x) \)

By pulling out the exponent, you transform a potentially difficult multiplication inside the log into a simpler multiplication outside.
In our exercise, after applying the product rule, we had \( \log_2(B^2) \). Using the Power Rule, this expression becomes \( 2 \log_2(B) \), because the exponent 2 moves outside of the logarithm.
Power Rule is powerful in expanding and simplifying logarithmic expressions. It is especially helpful when simplifying expressions for further calculation.

Logarithmic Expansion

Logarithmic Expansion is a technique used to expand and simplify logarithmic expressions. By using the laws of logarithms, specifically the product and power rules, complex expressions become more manageable.
In our given expression, \( \log_2(AB^2) \), applying these rules led us to fully expand the logarithm into \( \log_2(A) + 2 \log_2(B) \).
The goal of logarithmic expansion is to break down expressions into their simplest components. This makes them easier to work with in algebraic manipulations and problem-solving.
Expansion is often a crucial step in calculus for integrating or differentiating logarithmic functions, underscoring its significance beyond basic algebra.