Problem 13 - Algebra A Combined Function

Question

Write each difference as a single logarithm. Assume that variables represent positive numbers. $$ \log _{2} x-\log _{2} y $$

Step-by-Step Solution

Answer

\( \log_{2} \left( \frac{x}{y} \right) \)

Step 1: Identify the Formula for Logarithmic Differences

The expression is a difference of two logarithms: \( \log_{2} x - \log_{2} y \). The logarithmic property we use here is: \( \log_{b} a - \log_{b} c = \log_{b}\left( \frac{a}{c} \right) \). This property allows us to rewrite the difference as a single logarithm.

Step 2: Apply the Logarithmic Property

Apply the property identified in Step 1 to the expression: \( \log_{2} x - \log_{2} y = \log_{2} \left( \frac{x}{y} \right) \).

Step 3: Write the Final Expression

The final expression is now a single logarithm: \( \log_{2} \left( \frac{x}{y} \right) \). This represents the original difference between the two logarithms.

Key Concepts

Difference of Logarithms

When dealing with logarithms, a common operation you will come across is the "difference of logarithms". This operation is based on the subtraction of one logarithm from another, both having the same base. For instance, in the expression \( \log_{b} a - \log_{b} c \), both logarithms have a base \( b \).
The difference of logarithms has a unique property that allows it to be transformed into a different form. This transformation is key for simplifying log expressions and solving log equations.
This particular property states that:

• \( \log_{b} a - \log_{b} c = \log_{b} \left( \frac{a}{c} \right) \)
  Here, the difference of logarithms is equivalent to the logarithm of a quotient.

This conversion is not only mathematically elegant but also incredibly useful in various applications such as simplifying expressions or solving problems in exponential growth and decay scenarios.

Single Logarithm Expression

Transforming an expression involving multiple logarithms into a "single logarithm expression" is not only about simplification, but also about recognizing patterns and relations.
By converting a difference of logarithms into a single log, we use the concept of a quotient, making it simpler and often easier to solve.

• Using the property \( \log_{b} a - \log_{b} c = \log_{b} \left( \frac{a}{c} \right) \), we can condense the expression.

This property tells us that any expression of the form \( \log_{b} x - \log_{b} y \) can be rewritten as a single log expression \( \log_{b} \left( \frac{x}{y} \right) \).
This transformation reduces complexity and is a fundamental manipulation in logarithmic calculations, thus enabling its integration into more advanced algebraic processes or real-world applications.

Logarithmic Identities

"Logarithmic identities" are foundational formulas that help us manipulate and simplify logarithmic expressions. They are crucial when working with equations involving logarithms, allowing complex expressions to be simplified through known properties.

• The properties are derived from the rules of exponents because logarithms are essentially the inverse operations to exponentiation.


• Aside from the difference of logarithms, other frequent identities include:
  • Product of logs identity: \( \log_{b} a + \log_{b} c = \log_{b} (a \times c) \)
  • Power identity: \( \log_{b} (a^c) = c \cdot \log_{b} a \)
  • Change of base identity: \( \log_{b} a = \frac{\log_{k} a}{\log_{k} b} \) for any positive number \( k \)

These identities not only simplify expressions but also assist in solving logarithmic and exponential equations efficiently. By understanding and applying these basic principles, we gain a stronger grasp of how to handle logarithmic problems.