Problem 36 - Algebra 2 and Trigonometry

Question

Write each expression in terms of \(A\) and \(B\) if \(\log _{2} x=A\) and \(\log _{2} y=B\). \(\log _{2} x y\)

Step-by-Step Solution

Answer

The expression \( \log_{2} xy \) is equal to \( A + B \).

Step 1: Understand Key Identity of Logarithms

Recall the property of logarithms that states: \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \). This is known as the product rule for logarithms and can be very helpful in simplifying expressions involving products within logarithms.

Step 2: Apply the Product Rule

Given the expression \( \log_{2}(xy) \), apply the product rule from Step 1. This gives you \( \log_{2}(x) + \log_{2}(y) \).

Step 3: Substitute Given Values

From the problem, you know that \( \log_{2}(x) = A \) and \( \log_{2}(y) = B \). Substitute these into the expression from Step 2. So, \( \log_{2}(xy) = A + B \).

Key Concepts

Product Rule for Logarithms

The product rule for logarithms is a fundamental identity that helps simplify expressions where a logarithm is applied to a product of two numbers. It states that the logarithm of a product is equal to the sum of the logarithms of each individual factor. For any positive numbers \(M\) and \(N\) and a positive real number base \(b\), this identity is expressed as:

• \( \log_{b}(MN) = \log_{b}(M) + \log_{b}(N) \)

This property is immensely helpful when dealing with logarithmic expressions in algebra. It allows us to break down complex multiplications within a logarithm to simpler, more manageable additions. In the problem's example, it is used to transform \( \log_{2}(xy) \) into \( \log_{2}(x) + \log_{2}(y) \). This makes it easier to substitute the known values \(A\) and \(B\), which represent \( \log_{2}(x) \) and \( \log_{2}(y) \), respectively.

Properties of Logarithms

Logarithms come with several useful properties that make them valuable in mathematics, especially in solving equations involving exponential and logarithmic functions. Besides the product rule, two other important properties are:

• Quotient Rule: The logarithm of a quotient is the difference of the logarithms: \( \log_{b}(\frac{M}{N}) = \log_{b}(M) - \log_{b}(N) \).
• Power Rule: The logarithm of a number raised to a power is the exponent times the logarithm of the base number: \( \log_{b}(M^k) = k \cdot \log_{b}(M) \).

Each of these properties is aimed at simplifying the manipulation and transformation of logarithmic expressions. Recognizing when to apply these rules can greatly ease solving logarithmic equations. In our original exercise, understanding these properties, particularly the product rule, streamlined the process of expressing \( \log_{2}(xy) \) in terms of \(A\) and \(B\).

Algebraic Expressions

An algebraic expression is a combination of numbers and variables, along with arithmetic operations such as addition, subtraction, multiplication, and division. Algebraic expressions can represent a wide range of mathematical ideas from simple calculations to complex equations. In our context, understanding how logarithms fit into these expressions is crucial.The exercise involves expressing the algebraic form \( \log_{2}(xy) \) using simpler components \(A\) and \(B\). By using the product rule for logarithms, \( \log_{2}(xy) \) is restructured into an algebraic expression \(A + B\), showcasing how seemingly complex logarithmic forms can be simplified using algebraic principles.This transformation demonstrates the power of algebraic manipulation and reinforces the importance of understanding and applying logarithmic identities effectively in algebra.