Problem 3 - Algebra 2

Question

Write each expression as a single natural logarithm. \(\ln 24-\ln 6\)

Step-by-Step Solution

Answer

The expression \(\ln 24-\ln 6\) simplifies to \(\ln 4\).

Step 1: Identify Properties of Logarithms

The first step is identifying the properties of logarithms in use. In the given problem, \(\ln 24-\ln 6\), the property in use is the quotient property of logarithms. The quotient rule is a consequence of the rule of division for logarithms, \(\ln a - \ln b = \ln \frac{a}{b}\). Hence we can rewrite the expression using the quotient rule.

Step 2: Apply Quotient Rule

The next step is to apply the quotient property of logarithms. Using this property we can rewrite the expression as \(\ln \frac{24}{6}\).

Step 3: Simplify the Expresssion

The last step is simplifying the expression. Here we are dividing 24 by 6 which simplifies to 4. So the simplified expression would be \(\ln 4\).

Key Concepts

Properties of Logarithms

Logarithms have several key properties that help simplify complex expressions. One such property is the **quotient rule**, which is particularly useful when dealing with the difference of two logarithmic expressions. To fully understand this, let's revisit the basic idea of logarithms. A logarithm \( \ln a \) essentially answers the question, "To what exponent must the base (usually \( e \) for a natural logarithm) be raised, to result in \( a \)?"

Here's a short list of common properties of logarithms that you should keep in mind:

• Product Rule: \( \ln a + \ln b = \ln(ab) \)
• Quotient Rule: \( \ln a - \ln b = \ln \frac{a}{b} \)
• Power Rule: \( \ln(a^n) = n\ln(a) \)

Understanding these properties allows us to manipulate logarithmic expressions and combine them into simpler forms. In our exercise, identifying the use of these properties helps lay the groundwork for rewriting the expressions more concisely.

Quotient Rule

The quotient rule for logarithms is a powerful tool that allows us to transform the subtraction of two logarithms into a single logarithmic expression. This rule states that the logarithm of a division can be expressed as the difference between two logarithms:
\[ \ln a - \ln b = \ln \left( \frac{a}{b} \right) \]

By using the quotient rule, we can simplify and combine logarithmic expressions thereby making complex calculations more manageable. In the context of our exercise, when faced with \( \ln 24 - \ln 6 \), applying the quotient rule allows us to express this as \( \ln \frac{24}{6} \).

The goal here is to consolidate the expression, making it less cumbersome to work with and potentially leading to further simplifications through basic arithmetic.

Simplifying Expressions

Simplifying expressions is the process of reducing them to their simplest form. After applying the rules of logarithms, such as the quotient rule, further simplification often involves arithmetic operations like division or multiplication. In our example, once we rewrite \( \ln 24 - \ln 6 \) using the quotient rule to \( \ln \frac{24}{6} \), the next step is straightforward arithmetic.

Calculating \( \frac{24}{6} \) yields the result 4. Therefore, \( \ln \frac{24}{6} \) simplifies neatly to \( \ln 4 \).

Through simplification, we can often reach a cleaner, more concise version of our original expression. This makes it easier to interpret or use in further mathematical problems. The key is always to check if further simplification is possible, as it might uncover insights or simplifications not immediately visible in the original format.