Problem 9 - College Algebra with Corequisite Support

Question

For the following exercises, condense to a single logarithm if possible. \(\ln (7)+\ln (x)+\ln (y)\)

Step-by-Step Solution

Answer

The expression condenses to \( \ln(7xy) \).

Step 1: Recall the Logarithm Property

To condense multiple logarithms into a single logarithm, use the property of logarithms: \( \ln(a) + \ln(b) = \ln(a \cdot b) \). This property is known as the Product Rule for logarithms.

Step 2: Apply the Property to Two Terms

First, apply the Product Rule to the first two terms: \( \ln(7) + \ln(x) = \ln(7 \times x) \). Now the expression becomes \( \ln(7x) + \ln(y) \).

Step 3: Apply the Property to the Remaining Terms

Next, apply the Product Rule again to the resulting expression from Step 2: \( \ln(7x) + \ln(y) = \ln(7x \times y) \). This condenses all terms into a single logarithm.

Step 4: Write the Final Condensed Logarithm

The final condensed expression is \( \ln(7xy) \). This single logarithm represents the product of the original terms inside the logarithm.

Key Concepts

Product Rule

The product rule is a fundamental property of logarithms that's incredibly useful in mathematical operations. Simply put, the product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors. This can be expressed as:

• \( \ln(a) + \ln(b) = \ln(a \cdot b) \)

Imagine you have two numbers, say 3 and 4. According to the product rule, you don't have to compute \( \ln(3) \) and \( \ln(4) \) separately to understand \( \ln(12) \). Instead, just sum up \( \ln(3) + \ln(4) \) to reach \( \ln(12) \). This property saves time and offers a neat way to simplify expressions when dealing with multiple logarithmic terms.

Properties of Logarithms

Understanding the many properties of logarithms allows mathematicians to manipulate logarithmic expressions with ease. Besides the product rule, other key properties include:

• Quotient Rule: \( \ln \left( \frac{a}{b} \right) = \ln(a) - \ln(b) \)
• Power Rule: \( \ln(a^n) = n \cdot \ln(a) \)
• Change of Base Formula: \( \log_b(a) = \frac{\log_c(a)}{\log_c(b)} \) for any positive base \(c\)

These properties provide shortcuts to tackling logarithmic challenges. They help in expanding, simplifying, or changing the format of logarithmic expressions. In our exercise, the product rule was key, helping to condense multiple logs into one. However, similar strategies apply with these other properties for a variety of problems.

Condensing Logarithms

When presented with a lengthy logarithmic expression, you might have a headache looking at it. Instead, use the condensing technique to simplify it into a more compact form. Condensing is essentially the reverse of expanding—a method that combines terms using logarithmic properties.

• Identify which property can reduce the terms.
• Use the product, quotient, or power rules.
• Simplify step-by-step until you have one neat log expression.

For instance, in our original exercise, \( \ln(7) + \ln(x) + \ln(y) \) was transformed into a single log, \( \ln(7xy) \). This made it easier to interpret by showing a single factorized expression under the logarithm instead of multiple separated logs. Condensing enhances readability and aids practical applications, like solving equations or performing analytics.