Problem 41 - Algebra 2 and Trigonometry

Question

Write each expression in terms of \(A\) and \(B\) if \(\log _{2} x=A\) and \(\log _{2} y=B\). \(\log _{2}\left(x^{2} \div y^{3}\right)\)

Step-by-Step Solution

Answer

The expression is \(2A - 3B\).

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). Apply this rule to the expression \( \log_2(x^2 \div y^3) \) to separate it into two parts: \( \log_2(x^2) - \log_2(y^3) \).

Step 2: Apply the Power Rule to Each Logarithm

The power rule of logarithms states that \( \log_b(m^n) = n \cdot \log_b(m) \). Apply this to both terms: \( \log_2(x^2) = 2 \cdot \log_2(x) \) and \( \log_2(y^3) = 3 \cdot \log_2(y) \).

Step 3: Substitute Known Values for A and B

We know that \( \log_2(x) = A \) and \( \log_2(y) = B \). Substitute these into the expression: \( 2 \cdot A - 3 \cdot B \).

Step 4: Combine the Expression

After substituting, you are left with \( 2A - 3B \). This is the expression written in terms of \( A \) and \( B \).

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a valuable tool when dealing with logarithmic expressions that involve division. This rule tells us that the logarithm of a quotient can be transformed into the difference of two logarithms.
For a base, which we denote as \( b \), this rule is expressed as follows:

• \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \)

In simpler terms, when you divide \( m \) by \( n \), you can convert this division inside a logarithm into a subtraction operation between two distinct logarithms.
This transformation helps in simplifying and solving logarithmic expressions, making complex problems easier to tackle.
When applying the quotient rule to the given exercise \( \log_2(x^2 \div y^3) \), it gets simplified to \( \log_2(x^2) - \log_2(y^3) \).
Each component of the division becomes a separate term, allowing further manipulation.

Power Rule of Logarithms

The power rule of logarithms is another essential concept, particularly when dealing with expressions that involve exponents. It facilitates simplifying and working through logarithmic expressions by transforming powers into coefficients.
For any base \( b \), the power rule is represented by:

• \( \log_b(m^n) = n \cdot \log_b(m) \)

This equation tells us that when you have an exponent within a logarithm, you can "bring down" the power as a multiplier of the logarithm itself.
In the expression \( \log_2(x^2) - \log_2(y^3) \), the power rule allows us to rewrite it as:

• \( \log_2(x^2) = 2 \cdot \log_2(x) \)
• \( \log_2(y^3) = 3 \cdot \log_2(y) \)

Converting powers to coefficients significantly simplifies the expression because it turns a complex exponentiation into a straightforward multiplication.
It's crucial to remember that using this rule can transform otherwise daunting calculations into manageable steps.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions using algebraic rules. This skill is pivotal when transforming and solving mathematical expressions like logarithms.
Once you have applied the quotient and power rules to simplify the expression \( \log_2(x^2) - \log_2(y^3) \), the next step is to use algebraic manipulation to further simplify.
In the original problem, we substitute \( \log_2(x) = A \) and \( \log_2(y) = B \) into the expression to get the final form:

• \( 2 \cdot A - 3 \cdot B \)

Here, we replace the logarithmic terms with known variables to create an expression solely in terms of \( A \) and \( B \).
This manipulation turns an expression involving actual logarithmic values into something simpler and purely algebraic.
Hence, algebraic manipulation not only helps in simplifying expressions but also in making them easier to handle and understand. It's all about clear steps and logical transformations to streamline complex problems into straightforward solutions.