Problem 82 - Algebra 2

Question

Write each logarithmic expression as a single logarithm. $$ \log x-\log y $$

Step-by-Step Solution

Answer

The expression \(\log x-\log y\) can be rewritten as a single logarithm, \(\log (x/y)\).

Step 1: Identify the Property

The relevant property of logarithms is that \(\log_b (a) - \log_b (b) = \log_b (a/b)\). This subtractive property of logarithms allows us to reduce two separate logarithms of the same base into a singular logarithm.

Step 2: Apply the Property to the Expression

Now, apply this property to the logarithmic expression \(\log x-\log y\). Substitute the expressions in the form of the aforementioned property. Therefore, the equivalent expression would be \(\log (x/y)\). Here, the arguments x and y are divided because of the subtraction between the two logarithms.

Key Concepts

Logarithmic Expressions

Logarithmic expressions are mathematical statements that involve the use of logarithms. A logarithm answers the question: "To what exponent must a base number be raised, to produce a given number?" For example, in the expression \( \log_b(a) \), \( b \) is the base, and \( a \) is the number we're finding the logarithm for. This can be read as "log base \( b \) of \( a \)."

• Base is usually a positive number (excluding 1).
• The number \( a \) must always be positive because logarithms of zero or negative numbers aren't defined.

Logarithmic expressions come in various forms, including sums, differences, or products of logs, and often you will be asked to simplify them. This involves combining multiple logarithmic terms into a single one using specific properties of logarithms.

Logarithm Properties

There are several key properties of logarithms that are frequently used to simplify or manipulate logarithmic expressions. These properties help in transforming complex logarithmic expressions into simpler forms. Let’s explore some crucial properties:

• Product Property: \( \log_b(mn) = \log_b(m) + \log_b(n) \). This states that the log of a product is the sum of the logs.
• Quotient Property: \( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \). It tells us that the log of a quotient is the difference of the logs. This property was used in the original exercise.
• Power Property: \( \log_b(m^n) = n \cdot \log_b(m) \). This means the log of a power is the exponent times the log of the base number.

Understanding these properties is essential because they allow us to break down or simplify logarithmic expressions efficiently.

Simplifying Logarithms

Simplifying logarithms involves using the properties of logarithms to combine or reduce expressions. The goal is to express multiple logarithmic terms as a single term. Let's revisit the concept using the exercise as an example: \( \log x - \log y \).

• According to the Quotient Property of logarithms, the difference between two logs can be expressed as a single logarithm of a fraction: \( \log\left( \frac{x}{y} \right) \).

This process makes the expression neater and easier to interpret. Simplifying logarithmic expressions not only makes them more understandable but also solves complex problems more efficiently. Practice using properties and with time, simplifying logarithms will become second nature.