Problem 43 - Algebra A Combined Function

Question

Write each expression as a sum or difference of logarithms. Assume that variables represent positive numbers. $$ \log _{2} \frac{x^{3}}{y} $$

Step-by-Step Solution

Answer

The expression is written as: \( 3\log_2 x - \log_2 y \).

Step 1: Apply the Quotient Rule of Logarithms

The quotient rule for logarithms states that \( \log_b \frac{M}{N} = \log_b M - \log_b N \). So, apply this rule to the given expression: \( \log_2 \frac{x^3}{y} = \log_2 x^3 - \log_2 y \).

Step 2: Apply the Power Rule of Logarithms

The power rule for logarithms states that \( \log_b M^n = n \cdot \log_b M \). Apply this rule to the term \( \log_2 x^3 \) so that it becomes \( 3 \cdot \log_2 x \).

Step 3: Final Expression

Combine the expressions obtained from the previous steps to write the final expression: \( 3\log_2 x - \log_2 y \).

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is a fundamental principle that helps simplify expressions involving logarithms of fractions. Imagine you have an expression like \( \log_b \frac{M}{N} \). Here, \( M \) and \( N \) are both positive numbers. Using the quotient rule, this expression can be rewritten as \( \log_b M - \log_b N \).
This means you subtract the logarithm of the denominator from the logarithm of the numerator.

• This rule is useful because it breaks down complex expressions into simpler parts, making them easier to handle.
• Always ensure the bases of all logarithmic terms are the same for the rule to apply.

By understanding the quotient rule, you can confidently tackle problems involving fractions in logarithms and simplify them effectively.

Power Rule of Logarithms

The power rule of logarithms is another important tool that allows you to simplify expressions involving logarithms of powers. When you encounter an expression like \( \log_b M^n \), the power rule states that it can be transformed into \( n \cdot \log_b M \).
This transformation is basically taking the exponent in the expression and moving it in front of the logarithm as a multiplier.

• This approach makes it easier to manage powers within logarithmic expressions because it linearizes the exponent.
• Be aware that like other logarithmic rules, the power rule only works if \( M \) is positive and the base \( b \) remains consistent throughout the expression.

Mastering the power rule enables you to handle complex logarithmic expressions by simplifying them down to basic log components, which are much easier to compute or manipulate.

Logarithmic Identities

Logarithmic identities are powerful tools that help you rewrite and simplify logarithmic expressions. These identities include the various rules of logarithms, such as the quotient rule and the power rule, as well as others like the product rule.

• The product rule dictates that \( \log_b (MN) = \log_b M + \log_b N \), allowing you to separate the multiplication inside a log into a sum of two separate logs.
• These identities are very useful in solving and simplifying logarithmic equations since they help break apart and rearrange complex expressions into more manageable pieces.
• Focusing on these identities allows for a deeper understanding of logarithmic relationships, which is crucial when working with complex algebraic expressions in mathematics.

Understanding these logarithmic identities not only simplifies expressions but also equips you with the skill to handle a wide range of mathematical problems that include logarithms.